Evolutionary Equations
Christian Seifert Sascha Trostorff Marcus Waurick
math.AP math-ph math.CA math.FA math.MP 47D06, 35K, 35L, 35M, 35S, 35A, 35B35, 35F15, 35B27, 36Q61









23rd Internet Seminar

“Evolutionary Equations”
    Christian Seifert, Sascha Trostorff & Marcus Waurick
    Final Version, December 30, 2023
========================================================


















CHAPTER: INTRODUCTION


This chapter is intended to give a brief introduction as well as a
summary of the course to be presented throughout the semester. We
shall highlight some of the main ideas and methods behind the theory
and will also aim to provide some background on the main concept, which
will be the central object of study in the forthcoming weeks: the
notion of so-called

    Evolutionary Equations

dating back to Picard in the seminal paper <cit.>; see also
<cit.>. 

Another expression used to describe the same thing (and in order to
distinguish the concept from evolution equationevolution equations) is that of
evo-systems.evo-system Before going into detail on what we think of when
using the term evolutionary equations, we shall look into a seemingly
similar class of equations first.



§ EVOLUTION EQUATIONS


The term evolution equationevolution equation is commonly referred to as a (partial)
differential equation involving time. This is a well developed concept
that can be found, for example, in the standard references <cit.>.
Before addressing a solution strategy for these kinds of problems
we mention some examples. We shall revisit these examples again in
the course later. One of the main examples of evolution equationsevolution equation,
in the sense to be discussed in this section, is the heat equationheat equation
in its second order form. More precisely, 

    ∂_tθ(t,x)=Δθ(t,x),     (t,x)∈0∞×Ω,
    θ(0,x)=θ_0(x),     x∈Ω,

where Ω⊆^d is some open set, and Δ=∑_j=1^d∂_j^2
is the usual LaplacianLaplacian carried out with respect to the `x-variables' or `spatial variables',
and θ_0 is a given initial heat distribution and θ
is the unknown (scalar-valued) heat distribution. The above heat equation
is also accompanied with some boundary conditions for θ(t,x) which are required 
to be valid for all t>0 and x∈∂Ω.

We shall explain one way of solving this problem. To this end we make
a detour to the theory of ordinary differential equations. Let us consider
an n× n-matrix A with entries from the field  of complex
or real numbers,  or , and address the system of ordinary
differential equations 

    u'(t)=Au(t),     t>0,
    
    u(0)=u_0

for some given initial datum, u_0∈^n.
In this case, we know that there exists a unique solution. This solution can
be computed with the help of the so-called matrix exponentialmatrix exponential

    ^tA=∑_n=0^∞(tA)^k/k!∈^n× n

in the form 

    u(t)=^tAu_0.

As it turns out, this u is continuously differentiable and
 u satisfies the above equation. We note in particular that
 ^tAu_0→^0 Au_0=u_0 as t→0
and that ^(t+s)A=^tA^sA. In a way, to obtain
the solution for the system of ordinary differential equations we
need to construct (^tA)_t≥ 0. This is the same idea behind the process for obtaining a solution
for the aforementioned heat equation.

Indeed, given a suitable Banach space X one aims to construct a
so-called C_0-semigroupC_0-semigroup, (T(t))_t≥0, that
is, for all t≥ 0, T(t) is a bounded linear operator acting
in X, T(t)∈(X), and the following conditions are satisfied


  * semigroup law: T(0)= I and T(t+s)=T(t)T(s) for all t,s≥ 0,

  * strong continuity: for all x∈ X, lim_t→0T(t)x=x.

For instance in the case of X=Ł(Ω), it is possible to construct
such a family (T(t))_t≥0, written as (^tΔ)_t≥0,
satisfying the just mentioned criteria. For every θ_0∈Ł(Ω) this C_0-semigroup provides a function θ t↦^tΔθ_0∈Ł(Ω)
which satisfies the above heat equation in a certain generalised
sense. It is then an a posteriori question as to which additional conditions,
for example on θ_0, need to be imposed in order to assure that θ
solves the above heat equation as it stands. 

In the abstract setting of C_0-semigroups, the orbit
u t↦ T(t)x for some x∈ X then satisfies the equation

    u'(t)=Bu(t),     t>0,
    
    u(0)=x

provided that x∈y∈ XBylim_t→01/t(T(t)y-y)∈ X exists.
It can be shown that B is uniquely
determined. B is called the generatorgenerator of the C_0-semigroup
(T(t))_t≥ 0. The C_0-semigroup and the generator are in direct correspondence to
each other. In applications, given some operator B the task is
to find a C_0-semigroup such that Bx=lim_t→01/t(T(t)x-x)
for all x∈ X where either the left-hand side or the right-hand
side is well-defined. In other words, the question is whether a given
operator B is actually the generator of a C_0-semigroup.

Note that in the case of the heat equation, the C_0-semigroup
is also known as the fundamental solution or Green's functionfundamental solution or Green's function of the problem considered;
in the abstract setting, (T(t))_t≥0 is the fundamental
solution of eq:aCP.

As we have seen, C_0-semigroups focus on initial value problems. Moreover,
the heat equation (as a partial differential equation) above is viewed as an ordinary differential
equation with values in an infinite-dimensional state space
X. While the left-hand side of the equation is
always of the same form, the complexity of the problem class is stored
in the choice of X and the operator B (and, naturally, its domain of definition). 

In the literature, explicit initial value problems like that of the ODE system
or the heat equation are gathered under the umbrella term evolution
equation. It has become customary to refer to any problem where C_0-semigroups
play a fundamental role as an evolution equationevolution equation. This is particularly
the case when X is infinite-dimensional. Then, arguably, the study
of C_0-semigroups is the study of fundamental solutionsfundamental solution or Green's function (or abstract
Green's functions) associated to a certain class of initial
value problems for (partial) differential equations. A solution theory,solution theory, general notion
that is, the proof for existence, uniqueness and continuous dependence
on the data, is then contained in the construction of the fundamental
solution in terms of the ingredients of the equation. More precisely,
in the case of the ODE above, the fundamental solutionfundamental solution or Green's function is constructed
in terms of A and in case of the heat equation, the fundamental
solution is constructed in terms of (a particular realisation of)
Δ as an (unbounded) operator in X. We emphasise that there
is some bias towards the temporal direction in the theory of C_0-semigroups in the sense that C_0-semigroups impose continuity in time while this is in general not assumed for the spatial directions.



§ TIME-INDEPENDENT PROBLEMS


The construction of fundamental solutionsfundamental solution or Green's function is also a valuable method
for obtaining a solution for time-independent problems, see, e.g., <cit.>.
To see this, let us consider Poisson's equationPoisson's equation in ^3: Given
f∈(^3) we want to find a function u^3→
with the property that

    -Δ u(x)=f(x)  (x∈^3).

It can be shown that u given by

    u(x)=1/4π∫_^31/x-yf(y)ỵ

is well-defined, twice continuously differentiable and satisfies Poisson's
equation; cf. exer:Poisson_solution. Note that x↦1/4πx is also referred
to as the fundamental solution or Green's functionfundamental solution or Green's function
for Poisson's equation. The formula presented for u is the convolution
with the fundamental solution. The formula used
to define u also works for f being merely bounded and measurable
with compact support. In this case, however, the pointwise formula
of Poisson's equation cannot be expected to hold anymore, simply because
f is well-defined only up to a set of measure 0. Thus, only a posteriori
estimates under additional conditions on f render u to be twice
continuously differentiable (say) with Poisson's equation holding
for all x∈^3. However, similar to the semigroup setting,
it is possible to generalise the meaning of -Δ u=f.
Then, again, the fundamental solutionfundamental solution or Green's function can be used to construct a solution
for Poisson's equation for more general f. 

The situation becomes different when we consider a boundary value
problem instead of the problem above. More precisely, let Ω⊆^3
be an open set and let f∈Ł(Ω). We then need to ask whether
there exists u∈Ł(Ω) such that 

    -Δ u =f,     on Ω,
         u = 0,     on ∂Ω.

Notice that the task of just (mathematically) formulating this equation, let alone establishing a solution theory, is something that needs to be addressed. Indeed, we emphasise
that it is unclear as to what Δ u is supposed to mean if u∈Ł(Ω),
only. It turns out that the problem described is not well-posed in general.
In particular – depending on the shape of Ω and the norms
involved – it might, for instance, lack continuous dependence on the
data, f.

In any case, the solution formula that we have used for the case when Ω=^3
does not work anymore. Indeed, only particular shapes of Ω
permit to construct a fundamental solutionfundamental solution or Green's function; see <cit.>.
Despite this, when Ω is merely bounded, it is still possible
to construct a solution, u, for the above problem. There are two key ingredients required for this approach. One is a clever application of Riesz's representation
theorem for functionals in Hilbert spaces and the other one involves inventing `suitable'
interpretations of Δ u in Ω and u=0 on ∂Ω. Thus,
the method of `solving' Poisson's equation amounts to posing the
correct question, which then can be addressed without invoking
the fundamental solutionfundamental solution or Green's function. With this in mind, one could argue that the setting
makes the problem solvable. 



§ EVOLUTIONARY EQUATIONS


The central aim for evolutionary equationevolutionary equations is to combine the rationales
from both the C_0-semigroup theory and that from the time-independent
case. That is to say, we wish to establish a setting that treats time-independent
problems as well as time-dependent problems. At the
same time we need to generalise solution concepts. We
shall not aim to construct the fundamental solution in either
the spatial or the temporal directions. The problem class will comprise of problems that can be written in the form

    (∂_tM(∂_t)+A)U=F

where U is the unknown and F the known right-hand side.
Furthermore, A is an (unbounded, skew-selfadjoint) operator acting
in some Hilbert space that is thought of as modelling spatial coordinates;
∂_t is a realisation of the (time-)derivative operator and M(∂_t)
is an analytic, bounded operator-valued function M, which is evaluated at the time derivative.
In the course of the next chapters, we shall specfiy the definitions
and how standard problems fit into this problem class. 

Before going into greater depth on this approach, we would like to emphasise
the key differences and similarities which arise when compared to the derivation of more traditional 
solution theories that we outlined above.

Since the solution theory for evolutionary equationsevolutionary equation will also encapsulate
time-in­de­pen­dent problems, we cannot just focus on initial value problems
but rather on inhomogeneous problems. As we do not want to
require the existence of any fundamental solution we will also need to introduce
a generalisation of the concept of a solution. Indeed, continuity
in the form of the existence of a C_0-semigroup, or variants thereof,
will neither be shown nor be expected. Moreoever, we shall see that both
∂_t and A are unbounded operators with M(∂_t)
being a bounded operator. Thus, we need to make sense of the operator
sum of the two unbounded operators ∂_tM(∂_t)
and A, which, in general, cannot be realised as being onto but rather as
having dense range, only.

A post-processing procedure will then ensure that for more regular
right-hand sides, F, the solution U will also be more regular.
In some cases this will, for instance, amount to U being continuous
in the time variable. In this way, phrased in similar settings, C_0-semigroup
theory may be viewed as a regularity theory for a subclass of
evolutionary equationsevolutionary equation. We shall entirely confine ourselves within the
Hilbert space case though. In this sense, the solution theory to
be presented will be, in essence,
an application of the projection theorem (similar to time-independent problems). In our case, however, there
will not be as much of a regularity bias as there is in C_0-semigroup
theory or in abstract ODEs with an (infinite-)dimensional state space. In
fact, the projection theorem is applied in a Hilbert space, which
combines both spatial and temporal variables.

The operator M(∂_t) is thought of as carrying all the `complexity'
of the model. This is different to C_0-semigroups, where
this complexity is put on to the (domain of the) generator. What we mean by complexity  will become more
apparent when we discuss some examples. 

Finally, let us stress that A being `skew-selfadjoint' is a way
of implementing first order systems in our abstract setting. In fact,
deviating from classical approaches, we shall focus on first order
equations in both time and space. This is also another change
in perspective when compared to classical approaches. As classical treatments
might emphasise the importance of the Laplacian (and hence  Poisson's
equation) and variants thereof, evolutionary equationsevolutionary equation rather emphasise
Maxwell's equations as the prototypical PDE. This change of 
point of view will be illustrated in the following section, where we
address some classical examples.



§ PARTICULAR EXAMPLES AND THE CHANGE OF PERSPECTIVE


Here we will focus on three examples. These examples will also be the first to be readdressed when we
discuss the solution theory of evolutionary equationsevolutionary equation in a later chapter. In order to
simplify the current presentation we will not consider boundary value
problems but solely concentrate on problems posed on Ω=^3.
Furthermore, we shall dispose of any initial conditions. 



 §.§ Maxwell's Equations
Maxwell's equations

The prototypical evolutionary equation is the system provided by Maxwell's
equations. Maxwell's equations consist of two equations describing
an electro-magnetic field, (E,H), subject to a given certain external
current, J, 

    ∂_tε E+σ E- H     =-J,
    ∂_tμ H+ E     =0.

We shall detail the properties of the material parameters ε,μ, and σ
later on. For the time being it is safe to assume that they are non-negative
real numbers and that they additionally satisfy that μ(ε+σ)>0.
Now, in the setting of evolutionary equations, we gather the electro-magnetic
field into one column vector and obtain

    (∂_t[ ε 0; 0 μ ]+[ σ 0; 0 0 ]+[ 0 -;   0 ])[ E; H ]=[ -J;  0 ].

We shall see later that we obtain an evolutionary equation by setting

    M(∂_t)    [ ε 0; 0 μ ]+∂_t^-1[ σ 0; 0 0 ] and A[ 0 -;   0 ].


A formulation that fits well into the C_0-semigroup setting would
be, for example, 

    ∂_t[ E; H ]=[ ε 0; 0 μ ]^-1[ -σ   ;  -  0 ][ E; H ]+[ ε 0; 0 μ ]^-1[ -J;  0 ],

provided that ε>0. The inhomogeneous right-hand side (-1/ε J, 0)
can then be dealt with by means of the variation of constants formula,
which is the incarnation of the convolution of (-1/ε J, 0) with the fundamental solution
in this time-dependent situation. Thus, in order to apply semigroup
theory, the main task lies in showing that 

    [ -1/εσ   1/ε;  -1/μ     0 ]

is the generator of a C_0-semigroup. 

A different formulation needs to be put in place if ε=0.
The situation becomes even more complicated if ε and σ
are bounded, non-negative, measurable functions of the spatial variable such that ε+σ≥ c
for some c>0. In the setting of evolutionary equations, this problem,
however, can be dealt with. Note that then one cannot expect E to be continuous with respect to the temporal variable unless J is smooth enough.



 §.§ Wave Equation
wave equation

We shall discuss the scalar wave equation in a medium where the wave
propagation speed is inhomogeneous in different directions of space.
This is modelled by finding u×^3→ such that,
given a suitable forcing term f×^3→ (again
we skip initial values here), we have

    ∂_t^2u- a u=f,

where a=a^⊤∈^3×3 is positive definitepositive definite; that is, ξaξ_^3>0
for all ξ∈^3∖{0}. In the context of evolutionary
equations, we rewrite this as a first order problem in time and
space. For this, we introduce v∂_tu and q-a u
and obtain that

    (∂_t[    1    0;    0 a^-1 ]+[ 0  ;   0 ])[ v; q ]=[ f; 0 ].

Thus, 

    M(∂_t)[    1    0;    0 a^-1 ] and A[ 0  ;   0 ]

yield a corresponding formulation.
A semigroup formulation would work again by multiplying through
by [ 1 0; 0 a ] . 

Let us mention briefly that it is also possible to rewrite the wave
equation as a first order system in time only. For this, a standard
ODE trick is used: one simply sticks with the additional variable v=∂_tu
and obtains that

    ∂_t[ u; v ]=[ 0 1; a 0 ][ u; v ]+[ f; 0 ].

 In this formulation the `complexity' of the model is contained in the
operator 

    [ 0 1; a 0 ].

One would then have to show that this operator is the generator of a C_0-semigroup.



 §.§ Heat Equation
heat equation

We have already formulated the semigroup perspective of the heat equation

    ∂_tθ- aθ=Q,

in which we have added a heat source Q and a conductivity a=a^⊤∈^3×3
being positive definite. Here, again, we reformulate the heat equation as
a first order system in time and space to end up (again setting q-aθ) with 

    (∂_t[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[ 0  ;   0 ])[ θ; q ]=[ Q; 0 ].

In the context of evolutionary equations we then have that

    M(∂_t)[ 1 0; 0 0 ]+∂_t^-1[    0    0;    0 a^-1 ] and A[ 0  ;   0 ].

The advantage of this reformulation is that it becomes easily comparable
to the first order formulation of the wave equation outlined above.
For instance it is now possible to easily consider mixed type problems
of the form

    (∂_t[         1         0;         0 (1-s)a^-1 ]+[     0     0;     0 sa^-1 ]+[ 0  ;   0 ])[ θ; q ]=[ Q; 0 ],

with s^3→[0,1] being an arbitrary measurable function. In fact,
in the solution theory for evolutionary equations, this does not amount to any additional complication of the problem. Models of this type are particularly
interesting in the context of so-called solid-fluid interactionsolid-fluid interaction, where
the relations of a solid body and a flow of fluid surrounding it
are addressed.




§ A BRIEF OUTLINE OF THE COURSE


We now present an overview of the contents of the following chapters.



 §.§ Basics


In order to properly set the stage, we shall begin with
some background operator theory in Banach and Hilbert spaces. We
assume the readers to be acquainted with some knowledge on bounded
linear operators, such as the uniform boundedness principle, and basic
concepts in the topology of metric spaces, such as density and closure.
The most important new material will be the adjoint of an operator,
which need not be bounded anymore. In order to deal with this notion, we will
consider relations rather than operators as they provide the natural setting for unbounded operators. Having finished this brief detour
on operator theory, we will turn to a generalisation of Lebesgue spaces.
More precisely, we will survey ideas from Lebesgue's integration theory for functions
attaining values in an infinite-dimensional Banach space. 



 §.§ The Time Derivative


Banach space-valued (or rather Hilbert space-valued) integration theory
will play a fundamental role in defining the time derivative as
an unbounded, continuously invertible operator in a suitable Hilbert
space. In order to obtain continuous invertibility, we have to introduce
an exponential weighting function, which is akin to the exponential
weight introduced in the space of continuous functions for a proof of the Picard–Lindelf
theorem. It is therefore natural to discuss the application of
this operator to ordinary differential equations. In particular,
we will present a Hilbert space solution theory for ordinary differential
equations. Here, we will also have the opportunity to discuss ordinary differential
equations with delay and memory. After this short detour, we will
turn back to the time derivative operator and describe its spectrum.
For this we introduce the so-called Fourier–Laplace transformation
which transforms the time derivative into a multiplication operator.
This unitary transformation will additionally serve to define (analytic and
bounded) functions of the time derivative. This is absolutely essential for the
formulation of evolutionary equations.



 §.§ Evolutionary Equations


Having finished the necessary preliminary work, we will then be in a
position to provide the proper justification of the formulation and solution theory for evolutionary
equations. We will accompany this solution theory not only with the
three leading examples from above, but also with some more sophisticated
equations. Amazingly, the considered space-time setting will allow us to
discuss (time-)fractional differential equations, partial differential
equations with delay terms and even a class of integro-differential equations.
Withdrawing the focus on regularity with respect to the temporal variable,
we are en passant able to generalise well-posedness conditions from
the classical literature. However, we shall stick with the treatment of
analytic operator-valued functions M only. Therefore, we will
also include some arguments as to why this assumption seems to be physically
meaningful. It will turn out that analyticity and causality are intimately
related via both the so-called Paley–Wiener theorem and a representation
theorem for time translation invariant causal operators.



 §.§ Initial Value Problems for Evolutionary Equations


As it has been outlined above, the focus of evolutionary equations
is on inhomogeneous right-hand sides rather than on initial value
problems. However, there is also the possibility to treat initial value
problems with the approach discussed here. For this, we need to introduce
extrapolation spaces. This then enables us to formulate initial value
problems as inhomogeneous equations. We have to make a concession
on the structure of the problem, however. In fact, we will focus on the case when M(∂_t)=M_0+∂_t^-1M_1 for some bounded
linear operators M_0,M_1 acting in the spatial variables alone.
The initial condition will then read as (M_0U)(0)=M_0U_0.
Hence, one might argue that the initial condition U(0)=U_0 is
only assumed in a rather generalised sense. This is due to the fact
that M_0 might be zero. However, for the case A=0 we will
also discuss the initial condition U(0)=U_0, which amounts to
a treatment of so-called differential-algebraic equations in both
finite- and inifinite-dimensional state spaces. 



 §.§ Properties of Solutions and Inhomogeneous Boundary Value Problems


Turning back to the case when A≠0 we will discuss qualitative
properties of solutions of evolutionary equations. One of which will
be exponential decay. We will identify a subclass of evolutionary
equations where it is comparatively easy to show that if the right-hand side decays exponentially then so too must the solution. This is the
proper replacement in our setting for the notion of exponential stability
 from C_0-semigroups. If the right-hand side is smooth enough
we obtain that U(t), the solution of the evolutionary equation at time t,
decays exponentially if t→∞. Furthermore, we will frame
inhomogeneous boundary value problems in the setting of evolutionary
equations. The method will require a bit more on the regularity theory
for evolutionary equations and a definition of suitable boundary values.
In particular, we shall present a way of formulating classical inhomogeneous
boundary value problems for domains without any boundary regularity.



 §.§ Properties of the Solution Operator


In the final part, we shall have another
look at the advantages of the problem formulation. In fact, we will
have a look at the notion of homogenisation of differential equations.
In the problem formulation presented here, we shall analyse the continuity
properties of the solution operator with respect to weak operator topology convergence
of the operator M(∂_t). We will address an example for
ordinary differential equations (when A=0) and one for partial differential
equations (when A≠0). It will turn out that the respective continuity
properties are profoundly different from one another. 



§ COMMENTS


The focus presented here on the main notions behind evolutionary equations is
mostly in order to properly motivate the theory and highlight the
most striking differences in the philosophy. There are other solution
concepts (and corresponding general settings) developed for partial
differential equations; either time-dependent or without involving time. 

There is an abundance of examples and additional concepts for C_0-semigroups
for which we refer to the aforementioned standard treatments again.
There is also a generalisation to problems that are second order in
time, e.g., u”=Au, where u(0) and u'(0)
are given. This gives rise to cosine families of bounded linear operators which is
another way of generalising the fundamental solutionfundamental solution or Green's function concept, see,
for example, <cit.>.

The main focus of all of these equations is to address initial
value problems, where the (first/second) time derivative of the unknown
is explicit.

With a focus on static, that is, time-independent partial differential
equations, the notion of Friedrichs systems is also concerned with
a way of writing many PDEs from mathematical physics into a common
form, see <cit.>. A time-dependent
variant of constant coefficient Friedrichs systems are so-called symmetric-hyperbolic
systems, see e.g. <cit.>. In these cases, whether
the authors treat constant coefficients or not, the framework of evolutionary
equations adds a profound
amount of additional complexity by including the operator M(∂_t).

The treatment of time-dependent problems in space-time settings and
addressing corresponding well-posedness properties of a sum of two
unbounded operators has also been considered in <cit.>
with elaborate conditions on the operators involved. In their studies,
the flexibility introduced by the operator M(∂_t) in our
setting is missing, thus the time derivative operator is not
thought of having any variable coefficients attached to it.



§ EXERCISES

tocsectionExercises


Let ϕ∈ C(,). Assume that ϕ(t+s)=ϕ(t)ϕ(s) for
all t,s∈, ϕ(0)=1. Show that ϕ(t)=^α t (t∈)
for some α∈.



Let n∈, T→^n× n continuously differentiable such that
T(t+s)=T(t)T(s) for all t,s∈, T(0)=I. Show that there exists A∈^n× n
with the property that T(t)=^tA (t∈).




Show that x↦ u(x)=1/4π∫_^31/x-yf(y)ỵ
satisfies Poisson's equation, given f∈(^3).



Let f∈(). Define u(t,x) f(x+t) for x,t∈.
Show that u satisfies the differential equation ∂_tu=∂_xu
and u(0,x)=f(x) for all x∈. 



Let X,Y be Banach spaces, (T_n)_n∈ be a sequence in
(X,Y), the set of bounded linear operators. If supT_nn∈=∞,
show that there is x∈ X and a strictly increasing sequence (n_k)_k
in  such that T_n_kx→∞.



Let n∈. Denote by GL(n;) the set of continuously
invertible n× n matrices. Show that GL(n;)⊆^n× n
is open.



Let n∈. Show that ΦGL(n;)∋ A↦ A^-1∈^n× n
is continuously differentiable. Compute Φ'.


[heading=subbibliography]



CHAPTER: UNBOUNDED OPERATORS


We will gather some information on operators
in Banach and Hilbert spaces. Throughout this chapter let X_0, X_1, and
X_2 be Banach spaces and H_0, H_1, and H_2 be
Hilbert spaces over the field ∈{,}.



§ OPERATORS IN BANACH SPACES


The main difference of continuous linear operatorcontinuous linear operators,
that is, 

    (X_0,X_1)B X_0→ X_1B linear, Bsup_x∈ X_0∖{0}Bx/x<∞

(with the usual abbreviation (X_0):=(X_0,X_0))
and the set of uncontinuous or unbounded linear operators is that the latter only need to be defined 
on a subset of X_0. In order to define unbounded linear operators, we will first take a more general point of view and introduce (linear) relations.
This perspective will turn out to be the natural setting later on.




A subset A⊆ X_0× X_1 is called a relation
in X_0 and X_1. We define the domaindomain,
rangerange and kernelkernel of A
as follows

    (A)    x∈ X_0∃ y∈ X_1(x,y)∈ A,
    (A)    y∈ X_1∃ x∈ X_0(x,y)∈ A,
    (A)    x∈ X_0(x,0)∈ A.

The imageimage, A[M], of a set M⊆ X_0
under A is given by

    A[M]y∈ X_1∃ x∈ M(x,y)∈ A.

A relation A is called bounded, if for all bounded M⊆ X_0
the set A[M]⊆ X_1 is bounded. For a given relation A
we define the inverse relationinverse relation

    A^-1(y,x)∈ X_1× X_0(x,y)∈ A.

A relation A is called linear, if A⊆ X_0 × X_1 is a linear subspace. A linear
relation A is called linear operator or just operator
from X_0 to X_1, if 

    A[{0}] = y∈ X_1(0,y)∈ A={0}.

In this case, we also write 

    A(A)⊆ X_0→ X_1

to denote a linear operator from X_0 to X_1. Moreover, we
shall write Ax=y instead of (x,y)∈ A in this case.
A linear operator A, which is not bounded, is called unbounded.

For completeness, we also define the sum, scalar multiples, and composition
of relations.

Let A⊆ X_0× X_1, B⊆ X_0× X_1
and C⊆ X_1× X_2 be relations, λ∈𝕂. Then
we define

    A+B    (x,y+w)∈ X_0× X_1(x,y)∈ A,(x,w)∈ B,
    λ A    (x,λ y)∈ X_0× X_1(x,y)∈ A,
    
    CA    (x,z)∈ X_0× X_2∃ y∈ X_1(x,y)∈ A,(y,z)∈ C.



For a relation A⊆ X_0× X_1 we will use the abbreviation -A:=-1A (so that the minus sign only acts on the second component).
We now proceed with topological notions for relations.


Let A⊆ X_0× X_1 be a relation. A is called
densely defineddensely defined, if (A) is dense
in X_0. We call A closedclosed, if A is
a closed subset of the direct sum of the Banach spaces X_0 and
X_1. If A is a linear operator then we will call A closableclosable,
whenever A⊆ X_0× X_1 is a linear operator. 




Let A⊆ X_0× X_1 be a relation,
C∈(X_2,X_0) and B∈(X_0,X_1). Then the following
statements hold.


  * A is closed if and only if A^-1 is closed. Moreover, we have (A)^-1=A^-1.

  * A is closed if and only if A+B is closed;

  * if A is closed, then AC is closed.




Statement (a) follows upon realising that X_0× X_1∋(x,y)↦(y,x)∈ X_1× X_0
is an isomorphism. 

For statement (b), it suffices to show that
the closedness of A implies the same for A+B. Let ((x_n,y_n))_n
be a sequence in A+B convergent in X_0× X_1 to some (x,y). Since B∈(X_0,X_1), it follows that
((x_n,y_n-Bx_n))_n in A is convergent to
(x,y-Bx) in X_0× X_1. Since A is closed, (x,y-Bx)∈ A.
Thus, (x,y)∈ A+B.

For statement (c), let ((w_n,y_n))_n
be a sequence in AC convergent in X_2× X_1 to some (w,y). Since C is continuous, (Cw_n)_n
converges to Cw. Hence, (Cw_n,y_n)→(Cw,y) in X_0× X_1
and since (Cw_n,y_n)∈ A and A is closed, it follows that
(Cw,y)∈ A. Equivalently, (w,y)∈ AC, which yields closedness of AC.

We shall gather some other elementary facts about closed operators
in the following. We will make use of the following notion.


Let A(A)⊆ X_0→ X_1 be
a linear operator. Then the graph normgraph norm of A is defined by (A)∋ x↦x_A√(x^2+Ax^2).



Let A(A)⊆ X_0→ X_1 be
a linear operator. Then the following statements are equivalent:



  * A is closed.

  * (A) equipped with the graph norm
is a Banach space.

  * For all (x_n)_n in (A)
convergent in X_0 such that (Ax_n)_n is convergent in
X_1 we have lim_n→∞x_n∈(A) and Alim_n→∞x_n=lim_n→∞Ax_n.




For the equivalence (i)⇔(ii), it suffices to observe that (A)∋ x↦(x,Ax)∈ A,
where (A) is endowed with the graph norm, is an isomorphism.
The equivalence (ii)⇔(iii) is an easy reformulation of the definition of closedness of A⊆ X_0× X_1.


Unless explicitly stated otherwise (e.g. in the form (A)⊆ X_0, where we regard (A) as a subspace of X_0), for closed operators A we always consider (A) as a Banach
space in its own right; that is, we shall regard it as being endowed 
with the graph norm.


Let A(A)⊆ X_0→ X_1 be a
closed linear operator. Then A is bounded if and only if (A)⊆ X_0
is closed.



First of all note that boundedness of A is equivalent to the fact
that the graph norm and the X_0-norm on (A) are equivalent.
Hence, the closedness and boundedness of A implies that (A)⊆ X_0
is closed. On the other hand, the embedding 

    ι((A),·_A)↪((A),·_X_0)

 is continuous and bijective. Since the range is closed, the open
mapping theorem implies that ι^-1 is continuous. This yields
the equivalence of the graph norm and the X_0-norm and, thus,
the boundedness of A.


For unbounded operators, obtaining a precise description of the domain may be difficult. However, there may be a subset of the domain which essentially (or approximately) describes the operator.
This gives rise to the following notion of a core.


Let A⊆ X_0× X_1. A set D⊆(A) is
called a corecore for A provided A∩(D× X_1)=A.



Let A∈(X_0,X_1), and D⊆ X_0 a
dense linear subspace. Then D is a core for A.



Let A(A)⊆ X_0→ X_1
be a densely defined, bounded linear operator. Then there exists a
unique B∈ L(X_0,X_1) with B⊇ A. In particular,
we have B=A and 

    B=sup_x∈(A), x≠0Ax/x.


The proofs of prop:coreBounded and cor:uniquecont are asked for in exer:propcor.



§ OPERATORS IN HILBERT SPACES





Let us now focus on operators on Hilbert spaces. In this setting, we can additionally make use of scalar products ··, which in this course are considered to be linear in the second argument (and anti-linear in the first, in the case when =).

For a linear operator A(A)⊆ H_0→ H_1 the graph norm of A is induced
by the scalar product

    (x,y)↦xy+AxAy,

known as the graph scalar productgraph scalar product of A. If A is closed then (A) (equipped with the graph norm) is a Hilbert space.

Of course, no presentation of operators in Hilbert spaces would be complete without the central notion of the adjoint operator. We wish to pose the adjoint within the relational framework just established. The definition is as follows.

For a relation A⊆ H_0× H_1 we define the adjoint
relationadjoint relation A^* by

    A^*-((A^-1)^)⊆ H_1× H_0,

where the orthogonal complement is computed in the direct sum of the
Hilbert spaces H_1 and H_0; that is, the set H_1× H_0
endowed with the scalar product ((x,y),(u,v))↦xu_H_1+yv_H_0. 


Let A⊆ H_0× H_1. Then we have

    A^*=(u,v)∈ H_1× H_0∀(x,y)∈ A:uy_H_1=vx_H_0.

In particular, if A is a linear operator, we have

    A^*=(u,v)∈ H_1× H_0∀ x∈(A):uAx_H_1=vx_H_0.




Let A⊆ H_0× H_1 be a relation. Then A^*
is a linear relation. Moreover, we have

    A^*=-((A^)^-1)=((-A)^-1)^=(-(A^-1))^=((-A)^)^-1=(-(A^))^-1.



The proof of this lemma is left as exer:lem_adj.


Let A⊆ H_0× H_1. Since A^* is the orthogonal
complement of -A^-1, it follows immediately that A^* is closed. Moreover, A^* = (A)^* since A^ = (A)^.



Let A⊆ H_0× H_1 be a linear relation.
Then

    A^**(A^*)^*=A.




We compute using lem:adj

    A^**=((-(A^*))^-1)^=((-(-((A^)^-1)))^-1)^=(A^)^=A.*



Let A⊆ H_0× H_1 be a linear relation.
Then

    (A)^=(A^*)  and  (A^*)=(A)^.




Let u∈(A^*) and let y∈(A). Then we find x∈(A)
such that (x,y)∈ A. Moreover, note that (u,0)∈ A^*. Then, we
compute

    uy_H_1=0x_H_0=0.

This equality shows that (A)^⊇(A^*). If
on the other hand, u∈(A)^ then for all (x,y)∈ A
we have that

    0=uy_H_1,

which implies (u,0)∈ A^* and hence u∈(A^*). The
remaining equation follows from lem:ass together with
the first equation applied to A^*.

The following decomposition result is immediate from the latter theorem
and will be used frequently throughout the text.

 Let A⊆ H_0× H_1 be a closed linear relation.
Then






    H_1    =(A)⊕(A^*)  and  
    H_0 =(A)⊕(A^*).



We will now turn to the case where the adjoint relation is actually a linear operator.


Let A⊆ H_0× H_1 be a linear relation.
Then A^* is a linear operator if and only if A is densely
defined. If, in addition, A is a linear operator, then A is
closable if and only if A^* is densely defined.



For the first equivalence, it suffices to observe that 

    A^*[{0}]=(A)^.

Indeed, A being densely defined is equivalent to having (A)^={0}. Moreover, A^* is an operator if and only if
A^*[{0}]={0}. Next, we show eq:rel. For this, apply thm:ran-kernerl to the linear relation A^-1. One obtains ( A^-1)^=(A^-1)^*. Hence, ( (A))^ = (A^*)^-1=A^*[{0}], which is eq:rel.
For the remaining equivalence,
we need to characterise A being an operator. Using lem:ass
and the first equivalence, we deduce that A=(A^*)^*
is a linear operator if and only if A^* is densely defined.


 Note that the statement “A^* is an operator if  A is densely
defined”  asserted in lem:ddc is also true for any relation. For this, it suffices to observe that eq:rel is true for any relation A⊆ H_0 × H_1. Indeed, let A⊆ H_0× H_1 be a relation; define B A. Then (B)= (A). Also, we have 
    A^*=-(A^)^-1=-(B^)^-1=B^*.
 With these preparations, we can write

    (A)^    = ( (A))^
       = (B)^
       = B^*[{0}]
      = A^*[{0}],

where we used that eq:rel holds for linear relations.



Let A⊆ H_0× H_1 be a linear relation.
Then A∈(H_0,H_1) if and only if A^*∈(H_1,H_0). In either case, A^* = A.



Note that A∈ L(H_0,H_1) implies that A is closable
and densely defined. Thus, by lem:ddc, A^* is a
densely defined, closed linear operator. For u∈(A^*) we
compute using lem:ass

    A^*u=sup_x∈ H_0∖{0}A^*ux/x=sup_x∈ H_0∖{0}uAx/x≤Au,

yielding A^*≤A.
On the one hand, this implies that A^* is bounded, and on the other, since A^* is densely
defined we deduce A^*∈ L(H_1,H_0) by lem:bbc.
The other implication (and the other inequality) follows from the first one applied to A^*
instead of A using A^**=A.


We end this section by defining some special classes of relations and operators.


Let H be a Hilbert space and A⊆ H× H a linear relation.
We call A (skew-)Hermitian(skew-)Hermitian if
A⊆ A^* (A⊆ -A^*). We say that A is (skew-)symmetric(skew-)symmetric
if A is (skew-)Hermitian and densely defined (so that A^* is a linear operator), and A
is called (skew-)self­adjoint(skew-)selfadjoint if A=A^*
(A=-A^*). Additionally, if A is densely defined, then we say that
A is normalnormal if AA^*=A^*A. 




§ COMPUTING THE ADJOINT


In general it is a very difficult task to compute the adjoint of a
given (unbounded) operator. There are, however, cases, where the adjoint
of a sum or the product can be computed more readily. We start with the most basic
case of bounded linear operators.

Let A,B∈(H_0,H_1),C∈(H_2,H_0).
Then (A+B)^*=A^*+B^* and (AC)^*=C^*A^*.


The latter results are special cases of more general statements to
follow.

Let A⊆ H_0× H_1 be a relation and B∈ L(H_0,H_1).
Then (A+B)^*=A^*+B^*.



Let (u,v)∈ H_1× H_0. Then we compute

    (u,v)∈ A^*+B^*   (u,v-B^*u)∈ A^*
       ∀  (x,y)∈ Ayu_H_1=xv-B^*u_H_0
       ∀  (x,y)∈ Ay+Bxu_H_1=xv_H_0
       ∀  (x,z)∈ A+Bzu_H_1=xv_H_0
       (u,v)∈(A+B)^*.

Note that for the first, third and fourth equivalence, we have used the fact that
B∈(H_0,H_1) together with lem:bdds.


Let A⊆ H_0× H_1, B∈ L(H_0,H_1).
If A is densely defined, then A^*+B^* is an operator
and (A+B)^*=A^*+B^*.



Let A⊆ H_0× H_1 be a closed
linear relation and C∈(H_2,H_0). Then (AC)^*=C^*A^*.



We first show that AC⊆(C^*A^*)^*. For
this, let (w,y)∈ AC. Then (Cw,y)∈ A. Hence, for all (u,z)∈ C^*A^*;
that is, for all (u,v)∈ A^* and z=C^*v, we compute

    uy_H_1=vCw_H_0=C^*vw_H_2=zw_H_2,

which implies that AC⊆(C^*A^*)^*. Next, let
(w,y)∈(C^*A^*)^*. Then for all (u,v)∈ A^*
and z=C^*v we obtain

    uy_H_1=zw_H_2=C^*vw_H_2=vCw_H_0.

Thus, we obtain (Cw,y)∈ A^**=A=A. Thus, (w,y)∈ AC.
Hence, 

    AC=(C^*A^*)^*,

which yields the assertion by adjoining this equation.


Let A⊆ H_0× H_1 be a linear
relation and C∈(H_2,H_0). Then (AC)^*=C^*A^*.



The result follows upon realising that A^*=A^***=(A)^*.


Let A⊆ H_0× H_1 be a linear
relation and C∈(H_2,H_0). If AC is densely
defined, then C^*A^* is a closable linear operator with C^*A^*=(AC)^*.





 
  * Note that if B∈ L(H_1,H_2) and A⊆ H_0× H_1
linear, then (BA)^*=A^*B^*. Indeed,
this follows from thm:adj-comp applied to A^* and B instead
of A and C^*, respectively, since then we obtain (A^*B^*)^*=B^**A^**=BA.
Computing adjoints on both sides again and using that A^*B^*
is closed by prop:inv-sum-comp-closed, we get the assertion.


  * We note here that in cor:adj-comp1 and cor:adj-comp2 we AC cannot be replaced by AC
 and encourage the reader to find a counterexample for A being a
closable linear operator. We also refer to <cit.> for
a counterexample due to J. Epperlein.



We have already seen that A^*=A^*. We can even
restrict A to a core and still obtain the same adjoint.


Let A⊆ H_0× H_1 be a linear
relation, D⊆(A)  a linear subspace. Then D
is a core for A if and only if (A∩(D× H_1))^*=A^*.


 We set A|_D A∩ (D× H_1). Then

    D core   A|_D=AA|_D^=A^A|_D^=A^ A|_D^*=A^*.






§ THE SPECTRUM AND RESOLVENT SET


In this section, we focus on operators acting on a single Banach space. As such, throughout this section let X be a Banach space
over ∈{,} and let A(A)⊆ X→ X be
a closed linear operator.

The set 

    ρ(A)λ∈(λ-A)^-1∈(X)

is called the resolvent setresolvent set of A. We define

    σ(A)∖ρ(A)

to be the spectrumspectrum of A. 

We state and prove some elementary properties of the spectrum and
the resolvent set. We shall see natural examples for A which satisfy that
σ(A)= or σ(A)=∅ later on.

For a metric space (X,d), we will write xr=y∈ Xd(x,y)<r for the open ball around x of radius r and xr=y∈ Xd(x,y)≤ r for the closed ball.


If λ,μ∈ρ(A), then the resolvent identityresolvent
identity holds. That is

    (λ-A)^-1-(μ-A)^-1=(μ-λ)(λ-A)^-1(μ-A)^-1.

Moreover, the set ρ(A) is open. More precisely, if λ∈ρ(A)
then λ1/(λ-A)^-1⊆ρ(A)
and for μ∈λ1/(λ-A)^-1
we have

    (μ-A)^-1≤(λ-A)^-1/1-λ-μ(λ-A)^-1.


The mapping ρ(A)∋λ↦(λ-A)^-1∈(X)
is analytic.



For the first assertion, we let λ,μ∈ρ(A) and compute

    (λ-A)^-1-(μ-A)^-1    =(λ-A)^-1((μ-A)-(λ-A))(μ-A)^-1
        =(λ-A)^-1(μ-λ)(μ-A)^-1
        =(μ-λ)(λ-A)^-1(μ-A)^-1.

Next, let λ∈ρ(A) and μ∈λ1/(λ-A)^-1.
Then 

    (λ-μ)(λ-A)^-1<1.

Hence, 1-(λ-μ)(λ-A)^-1 admits
an inverse in (X) satisfying

    (1-(λ-μ)(λ-A)^-1)^-1=∑_k=0^∞((λ-μ)(λ-A)^-1)^k.

We claim that μ∈ρ(A). For this, we compute

    μ-A     =λ-A-(λ-μ)
     =(λ-A)(1-(λ-μ)(λ-A)^-1).

Since (1-(λ-μ)(λ-A)^-1) is
an isomorphism in (X), we deduce that the right-hand side admits
a continuous inverse if and only if the left-hand side does. As λ∈ρ(A),
we thus infer μ∈ρ(A). The estimate follows from eq:Neu.
Indeed, we have

    (μ-A)^-1   ≤(λ-A)^-1∑_k=0^∞((λ-μ)(λ-A)^-1)^k
       ≤(λ-A)^-1∑_k=0^∞(λ-μ)(λ-A)^-1^k
     =(λ-A)^-1/1-(λ-μ)(λ-A)^-1.


For the final claim of the present proposition, we observe that 

    (μ-A)^-1    =(1-(λ-μ)(λ-A)^-1)^-1(λ-A)^-1
     =∑_k=0^∞(λ-μ)^k((λ-A)^-1)^k+1,

which is an operator norm convergent power series expression for the
resolvent at μ about λ. Thus, analyticity follows.

We shall consider multiplication operators in Ł(μ) next. For
a measurable function VΩ→ we will use the notation
[V≤ c] V^-1[-∞c] for some
constant c∈ℝ (and similarly for relational symbols other
than ≤).

Let (Ω,Σ,μ) be a measure
space and VΩ→ a measurable function. Then the operator

    V()(V())⊆Ł(μ)    →Ł(μ)
    
    f    ↦(ω↦ V(ω)f(ω)),
 
with (V())f∈Ł(μ)(ω↦ V(ω)f(ω))∈Ł(μ) satisfies the following prop­er­ties:

  
  * V() is densely defined and closed.
  
  * (V())^*=V^*() where V^*(ω)=V(ω)^*
for all ω∈Ω (here V(ω)^* denotes the complex
conjugate of V(ω)).

  * If V is μ-almost everywhere bounded,
then V() is continuous. Moreover, we have V()_(Ł(μ))≤V_L_∞(μ).

  * If V≠0 μ-a.e. then V() is
injective and V()^-1=1/V(), where 

    1/V(ω)1/V(ω),     V(ω)≠0,
    
    0,     V(ω)=0,

for all ω∈Ω. 
  


For the whole proof we let Ω_n[V≤ n] and put _n_Ω_n.  

(a) We first show that V() is densely defined.
Let f∈Ł(μ). Then, we have for all n∈ that _nf∈(V()).
From Ω=⋃_nΩ_n and Ω_n⊆Ω_n+1
it follows that _nf→ f in Ł(μ) as n→∞.

Next, we confirm that V() is closed. Let (f_k)_k in (V())
convergent in Ł(μ) with (V()f_k)_k be convergent in Ł(μ).
Denote the respective limits by f and g. It is clear that for
all n∈ we have _nf_k→_nf as k→∞. Also,
we have 

    _ng=lim_k→∞_nV()f_k=lim_k→∞V()(_nf_k)=V()(_nf) = _nVf.

Hence, g=Vf μ-almost everywhere and since g∈Ł(μ),
we have that f∈(V()). 

(b) It is easy to see that V^*()⊆ V()^*. For the other inclusion, we
let u∈(V()^*). Then, for all f∈Ł(μ) and n∈ we have _n f∈(V()) and, hence,

    f_nV^*u = ∫_Ω_n f^* V^* u μ̣= V()(_nf)u=_nfV()^*u=f_nV()^*u.

It follows that _nV^*u=_nV()^*u for all n∈.  Thus, Ω=⋃_nΩ_n implies V^*u=V()^*u and therefore u∈(V^*()) and V^*()u=V()^*u.

(c) If V≤κ μ-almost everywhere for some κ≥0,
then for all f∈Ł(μ) we have V(ω)f(ω)≤κf(ω)
for μ-almost every ω∈Ω. Squaring and integrating this
inequality yields boundedness of V() and the asserted inequality.

(d) Assume that V≠0 μ-a.e. and V()f=0. Then, f(ω)=0
for μ-a.e. ω∈Ω, which implies f=0 in Ł(μ).
Moreover, if V()f=g for f,g∈Ł(μ), then for μ-a.e. ω∈Ω we deduce that f(ω)=1/V(ω)g(ω),
which shows 1/V()⊇ V()^-1. If on the other
hand g∈(1/V()), then a similar computation
reveals that 1/V()g∈(V()) and V()1/V()g=g. 


The spectrum of V() from the latter example can be computed once
we consider a less general class of measure spaces. For later use, we provide a characterisation of these measure spaces first.

 Let (Ω,Σ,μ) be a measure
space.
Then the following statements are equivalent:


  
  * (Ω,Σ,μ) is semi-finitesemi-finite, that is, for every A∈Σ with μ(A)=∞,
there exists B∈Σ with 0<μ(B)<∞ such that B⊆ A.
  
  * For all measurable VΩ→ with V()∈(Ł(μ)), we have V∈ L_∞(μ) and V_L_∞(μ)≤V()_(Ł(μ)).



 (i)⇒(ii): Let ε>0 and A_ε [V≥V()_(Ł(μ))+ε]. Assume that μ(A_ε)>0. Since (Ω,Σ,μ) is semi-finite we find B_ε⊆ A_ε such that 0<μ(B_ε)<∞. Define fμ(B_ε)^-1/2_B_ε∈Ł(μ) with f_Ł(μ)=1. Consequently, we obtain
 
    V()_(Ł(μ))≥V()f_Ł(μ)≥V()_(Ł(μ))+ε,
which yields a contradiction, and hence (ii).
 
 (ii)⇒(i): Assume that (Ω,Σ,μ) is not semi-finite. Then we find A∈Σ with μ(A)=∞ such that for each B⊆ A measurable, we have μ(B)∈{0,∞}. Then V_A is bounded and measurable with V_L_∞(μ)=1. However, V()=0. Indeed, if f∈Ł(μ) then [f≠ 0]=⋃_n∈[f^2≥ n^-1]. Thus, 
 
    [V()f≠ 0] = [f≠ 0]∩ A = ⋃_n∈ [f^2≥ n^-1]∩ A.
 Since μ([f^2≥ n^-1])<∞ as f∈Ł(μ), we infer μ([f^2≥ n^-1]∩ A)=0 by the property assumed for A. Thus, μ([V()f≠ 0])=0 implying V()=0. Hence, V()_(Ł(μ))=0<1=V_L_∞(μ).
 

A straightforward consequence of thm:mult1 (c) and prop:mult1.5 is the following.

 Let (Ω,Σ,μ) be a semi-finite measure
space, VΩ→ measurable and bounded. Then V()_(Ł(μ))=V_L_∞(μ).



Let (Ω,Σ,μ) be a semi-finite measure
space and let VΩ→ be measurable. Then

    σ(V())= Vλ∈∀ε>0μ([λ-V<ε])>0.



Let λ∈ V. For all n∈ we find B_n∈Σ
with non-zero, but finite measure such that B_n⊆[λ-V<1/n].
We define f_n√(1/μ(B_n))_B_n∈Ł(μ).
Then f_n_Ł(μ)=1 and 

    V(ω)f_n(ω)≤V(ω)-λf_n(ω)+λf_n(ω)≤(1/n+λ)f_n(ω)

for ω∈Ω, which shows that (f_n)_n is in (V()). A similar estimate, on the other hand, shows that

    (V()-λ)f_n_Ł(μ)→0  (n→∞).

Thus, (V()-λ)^-1 cannot be continuous as f_n_Ł(μ)=1
for all n∈ℕ. 

Let now λ∈∖ V. Then there exists ε>0
such that N[λ-V<ε] is
a μ-null set. In particular, λ-V≠0 μ-a.e. Hence,
(λ-V())^-1=1/λ-V() is a linear
operator. Since, 1/λ-V≤1/ε μ-almost
everywhere, we deduce that (λ-V())^-1∈(Ł(μ))
and hence, λ∈ρ(V()). 


We conclude this chapter by stating that multiplication operators
as discussed in thm:mult1, prop:mult1.5, prop:mult1.75, and thm:mult2
are the prototypical example for normal operators. It is also important to note that, as we have seen in thm:mult1,
 a multiplication operator in Ł(μ) is self-adjoint if and
only if V assumes values in the real numbers, only.



§ COMMENTS


The material presented in this lecture is basic textbook knowledge.
We shall thus refer to the monographs <cit.>.
Note that spectral theory for self-adjoint operators is a classical
topic in functional analysis. For a glimpse on further theory of linear
relations we exemplarily refer to <cit.>. 
The restriction in prop:mult1.75 and thm:mult2 to semi-finite measure spaces is not very severe. In fact, if (Ω,Σ,μ) was not semi-finite, it is possible to construct a semi-finite measure space (Ω_loc,Σ_loc,μ_loc) such that L_p(μ) is isometrically isomorphic to L_p(μ_loc), see <cit.>.



§ EXERCISES

tocsectionExercises


Let A⊆ X_0× X_1 be an unbounded linear operator. 
Show that for every linear operator B⊆ X_0× X_1
with B⊇ A and (B)=X_0, we have that B is not
closed.



Prove prop:coreBounded and cor:uniquecont. Hint: One might use that bounded linear relations are always operators.



Prove lem:adj.



Let A(A)⊆ H_0→ H_0 be a closed and densely
defined linear operator. Show that for all λ∈ we have

    λ∈ρ(A)λ^*∈ρ(A^*).




Let U⊆ H_0× H_1 satisfy U^-1=U^*. Show
that U∈ L(H_0,H_1) and that U is unitaryunitary,
that is, U is onto and for all x∈ H_0 we have Ux_H_1=x_H_0.



Let δ C01⊆Ł(0,1)→,f↦ f(0),
where C01 denotes the set of -valued continuous functions
on [0,1]. Show that δ is not closable. Compute δ.



Let C⊆ be closed. Provide a Hilbert space H and a densely
defined closed linear operator A on H such that σ(A)=C.


  [heading=subbibliography]
  
  

CHAPTER: THE TIME DERIVATIVE


It is the aim of this chapter to define a derivative operator on a
suitable Ł-space, which will be used as the derivative with respect
to the temporal variable in our applications. As we want to deal with
Hilbert space-valued functions, we start by introducing the concept
of Bochner–Lebesgue spaces, which generalises the classical scalar-valued
L_p-spaces to the Banach space-valued case. 



§ BOCHNER–LEBESGUE SPACES


Throughout, let (Ω,Σ,μ) be a σ-finite measure space and X
a Banach space over the field 𝕂∈{,}. We are aiming
to define the spaces L_p(μ;X) for 1≤ p≤∞. This is the
space of (equivalence classes of) measurable functions attaining values
in X, which are p-integrable (if p<∞), or essentially
bounded (if p=∞) with respect to the measure μ. We begin
by defining the space of simple functions on Ω with values
in X and the notion of Bochner-measurability.

For a function fΩ→ X and x∈ X we set 

    A_f,x f^-1[{x}].

A function fΩ→ X is called simplesimple function
if f[Ω] is finite and for each x∈ X∖{0} the
set A_f,x belongs to Σ and has finite measure. We denote
the set of simple functions by S(μ;X).
A function fΩ→ X is called Bochner-measurableBochner-measurable
if there exists a sequence (f_n)_n∈ in S(μ;X) such
that 

    f_n(ω)→ f(ω)  (n→∞)

for μ-a.e. ω∈Ω.

 



  * For a simple function f we have 

    f=∑_x∈ X x·_A_f,x,

where the sum is actually finite, since _A_f,x=0 for all x∉ f[Ω].

  *  If X=, then a function is Bochner-measurable if and only if it has a μ-measurable representative.

  * It is easy to check that S(μ;X) is a vector space and an S(μ;)-module;
that is, for f∈ S(μ;X) and g∈ S(μ;) we have 
g· f∈ S(μ;X).


  *  If fΩ→ X is Bochner-measurable, then f(·)_XΩ→
is Bochner-measurable. Indeed, since 

    f(·)_X=lim_n→∞f_n(·)_X

for μ-a.e. ω∈Ω and a sequence (f_n)_n∈
in S(μ;X), it suffices to show that f_n(·)_X
is simple for all n∈. The latter follows since A_f,x∩ A_f,y=∅
for x y and thus

    f_n(·)_X=∑_x∈ f_n[Ω]x_X·_A_f,x

is a real-valued simple function.


  * If one deals with arbitrary measure spaces, the definition of simple functions has to be weakened by allowing the sets A_f,x to have infinite measure. However, since in the applications to follow we only work with weighted Lebesgue measures, we restrict ourselves to σ-finite measure spaces. 



Next, we state and prove the celebrated Theorem of Pettis, which characterises Bochner-measurabilty in terms of weak measurability. In what follows, let X':=(X,)dual space X' denote the dual space of X.

 
 Let f:Ω→ X.
Then f is Bochner-measurable if and only if


  * f is weakly Bochner-measurableweakly Bochner-measurable; that is, x'∘ f:Ω→𝕂
is Bochner-measurable for each x'∈ X', and

  * f is almost separably-valuedalmost separably-valued; that is, f[Ω∖ N_0]
is separable for some N_0∈Σ with μ(N_0)=0. 




If f is Bochner-measurable, then clearly it is weakly Bochner-measurable.
Further, as f is the almost everywhere limit of simple functions,
it is almost separably-valued, since each simple function attains
values in a finite dimensional subspace of X.

Assume now conversely that f satisfies (a) and (b). We define Y f[Ω∖ N_0],
which is a separable Banach space by (b). Thus, there exists a sequence
(x_n')_n∈ℕ in X' such that 

    y=sup_n∈ℕ|x_n'(y)|  (y∈ Y).

Since for each n∈ the function g_n|x_n'∘ f|
is Bochner-measurable by (a) and rem:Bochner<ref>, we find a μ-nullset N_n and
a measurable function g̃_n:Ω→ such that g_n=g̃_n
on Ω∖ N_n by rem:Bochner<ref>. Then sup_n∈g̃_n(·)
is measurable and 

    f(ω)=sup_n∈g̃_n(ω)  (ω∈Ω∖ N),

where N⋃_n∈_0N_n, which shows that f(·)
is Bochner-measurable. Let ε>0, (y_n)_n∈ℕ
a dense sequence in Y. Applying the previous argument to the function
f_k(·) f(·)-y_k for k∈ we infer that
f_k(·) is Bochner measurable and hence, there is a μ-nullset
N_k' and a measurable funtion f̃_̃k̃:Ω→ such
that f_k=f̃_k on Ω∖ N_k'. Consequently,
the sets 

    E_k[f̃_k≤ε]ω∈Ω|f̃_k(ω)|≤ε  (k∈ℕ)

are measurable. Moreover, by the density of {y_n ; n∈}
in Y, we get that Ω∖ N'⊆⋃_k∈E_k
with N'⋃_k=1^∞N_k'∪ N_0. Setting
F_1 E_1 and F_n+1=E_n+1∖⋃_k=1^nF_k
for n∈ℕ, we obtain a sequence of pairwise disjoint measurable
sets (F_n)_n∈ℕ with Ω∖ N'⊆⋃_n∈ℕF_n.
We set 

    g∑_k=1^∞y_k_F_k

and obtain f(ω)-g(ω)≤ε for each ω∈Ω∖ N'.
Hence, if g is Bochner-measurable, then f is Bochner-measurable as well. For showing the Bochner-measurability of g, let (Ω_k)_k∈ℕ
be a sequence of pairwise disjoint measurable sets such that ⋃_k∈ℕΩ_k=Ω
and μ(Ω_k)<∞ for each k∈ℕ. For n∈ℕ
we set 

    g_n∑_k,j=1^ny_k_F_k∩Ω_j.

Then (g_n)_n∈ℕ is a sequence of simple
functions with g_n→ g pointwise as n→∞ and thus,
g is Bochner-measurable.




 Let f_n,f:Ω→ X for n∈. Moreover, assume that f_n is Bochner-measurbale for each n∈ and f_n(ω)→ f(ω) as n→∞ for μ-almost every ω∈Ω. Then f is Bochner-measurable 



 By thm:Pettis we find μ-nullsets N_n∈Σ such that X_n f_n[Ω∖ N_n] is separable. Moreover, we find a μ-nullset N'∈Σ such that 
 
    f_n(ω)→ f(ω)   (n→∞, ω∈Ω∖ N').

We set N⋃_n∈ N_n∪ N'. Then 

    f[Ω∖ N]⊆⋃_n∈ X_n

and thus, f is almost separably-valued. Moreover, for x'∈ X' we have that 

    (x'∘ f)(ω)=lim_n→∞ (x'∘ f_n)(ω)   (ω∈Ω∖ N')

and since all functions x'∘ f_n are μ-measurable outside a μ-nullset, so is x'∘ f and thus, x'∘ f is Bochner-measurable, see rem:Bochner<ref>. Thus, thm:Pettis yields the Bochner-measurability of f.


[Bochner–Lebesgue spaces]
 Bochner-Lebesgue spacesFor p∈1∞ we define

    ℒ_p(μ;X)fΩ→ Xf, f(·)_X∈ℒ_p(μ),

as well as 

    L_p(μ;X)ℒ_p(μ;X)∼,

where ∼ denotes the usual equivalence relation of equality μ-almost
everywhere.


Let p∈1∞. Then 

    f_p(_Ωf(ω)_X^pμ̣(ω))^1/p,     if p<∞,
    _ω∈Ωf(ω)_X,     if p=∞,

defines a seminorm on ℒ_p(μ;X) where f_p=0
if and only if f=0 μ-a.e. Consequently, ·_p
defines a norm on L_p(μ;X). Moreover, (L_p(μ;X),·_p)
is a Banach space and if X=H is a Hilbert space, then so too is L_2(μ;H)
with the scalar product given by 

    fg_2_Ωf(ω)g(ω)_Hμ̣(ω)  (f,g∈ L_2(μ;H)).




We just show the completeness of L_p(μ;X). Let (f_n)_n∈
be a sequence in L_p(μ;X) such that ∑_n=1^∞f_n_p<∞.
We set 

    g_n(ω)f_n(ω)_X  (n∈,ω∈Ω).

Then (g_n)_n∈ is a sequence in L_p(μ) such that
∑_n=1^∞g_n_p<∞. By the completeness
of L_p(μ) we infer that 

    g∑_n=1^∞g_n

exists and is an element in L_p(μ). In particular, g(ω)<∞
for μ-a.e. ω∈Ω and thus, 

    ∑_n=1^∞f_n(ω)_X=∑_n=1^∞g_n(ω)<∞

for μ-a.e. ω∈Ω. By the completeness of
X we can define 

    f(ω)∑_n=1^∞f_n(ω)

for μ-a.e. ω∈Ω. Note that f is Bochner-measurable by cor:limit-measurable. We need to prove that
f∈ L_p(μ;X) and that ∑_n=1^kf_n→ f in L_p(μ;X)
as k→∞.  For this, it suffices to prove that

    ∑_n=k^∞f_n∈ L_p(μ;X)∑_n=k^∞f_n→0L_p(μ;X)k→∞.

Indeed, this would imply both that f-∑_n=1^kf_n∈ L_p(μ;X)
and the desired convergence result. We prove eq:series_to_zero
for p<∞ and p=∞ separately.

First, let p=∞. For each n∈ we have f_n∈ L_∞(μ;X)
and thus f_n(ω)_X≤f_n_∞ for
all ω∈Ω∖ N_n and some null set N_n⊆Ω. We set N⋃_n=1^∞N_n,
which is again a null set. For k∈ and ω∈Ω∖ N
we then estimate 

    ∑_n=k^∞f_n(ω)_X≤∑_n=k^∞f_n(ω)_X≤∑_n=k^∞f_n_∞,

which yields eq:series_to_zero.

Now, let p<∞. For k∈ we estimate 

    (_Ω(∑_n=k^∞f_n(ω)_X)^pμ̣(ω))^1/p   ≤(_Ω(∑_n=k^∞f_n(ω)_X)^pμ̣(ω))^1/p
        =(_Ωlim_m→∞(∑_n=k^mf_n(ω)_X)^pμ̣(ω))^1/p
        =lim_m→∞(_Ω(∑_n=k^mf_n(ω)_X)^pμ̣(ω))^1/p
       ≤lim_m→∞∑_n=k^mf_n_p
     =∑_n=k^∞f_n_p,

where we have used monotone convergence in the third line. This estimate
yields eq:series_to_zero.


We now want to define an X-valued integral for functions in L_1(μ;X);
the so-called Bochner-integral. To do this, we need the following
density result.

The space S(μ;X) is dense in L_p(μ;X)
for p∈1∞.



Let f∈ L_p(μ;X). Then there exists a sequence (f_n)_n∈
in S(μ;X) such that f_n(ω)→ f(ω) for all ω∈Ω∖ N
for some null set N⊆Ω. W.l.o.g. we may assume that f_n(·)_X and f(·)_X are measurable on Ω∖ N for each n∈. For
n∈ we define the set 

    I_nω∈Ω∖ Nf_n(ω)_X≤2f(ω)_X∈Σ,

and set f̃_n f_n_I_n. Then f̃_n∈ S(μ;X)
and we claim that f̃_n(ω)→ f(ω) for all ω∈Ω∖ N.
Indeed, if f(ω)=0 then f̃_n(ω)=0 also and
the claim follows. If f(ω)0, then there is some n_0∈
such that f_n(ω)_X≤2f(ω)_X for
n≥ n_0, and hence ω∈⋂_n≥ n_0I_n. Consequently
f̃_n(ω)=f_n(ω)→ f(ω). By dominated
convergence, it now follows that 

    _Ωf̃_n(ω)-f(ω)_X^pμ̣(ω)→0  (n→∞),

which proves the claim. 


The mapping[Note that the sum is indeed finite and all summands are well-defined
if we set 0_X·∞=0_X.] 

    ∫_Ωμ̣ S(μ;X)⊆ L_1(μ;X)    → X
    
    f    ↦∑_x∈ X x·μ(A_f,x)

is linear and continuous, and thus has a unique continuous linear extension
to L_1(μ;X), called the Bochner-integralBochner-integral. Moreover,

    ∫_Ωfμ̣_X≤f_1  (f∈ L_1(μ;X)),

and for A∈Σ,f∈ L_1(μ;X) we set 

    ∫_Afμ̣∫_Ωf·_Aμ̣.

 



We first show linearity. Let f,g∈ S(μ;X) and λ∈. We then compute

    ∫_Ω(λ f+g)μ̣    =∑_x∈ Xx·μ(A_λ f+g,x)
     =∑_x∈ Xx·μ(⋃_y∈ X(A_f,y∩ A_g,x-λ y))
        =∑_x∈ X∑_y∈ Xx·μ(A_f,y∩ A_g,x-λ y)
        =∑_y∈ X∑_x∈ Xλ y·μ(A_f,y∩ A_g,x-λ y)+∑_x∈ X∑_y∈ X(x-λ y)μ(A_f,y∩ A_g,x-λ y)
        =λ∑_y∈ Xy·μ(A_f,y∩⋃_x∈ XA_g,x-λ y)+∑_z∈ Xz·μ(⋃_y∈ XA_f,y∩ A_g,z)
        =λ∫_Ωfμ̣+∫_Ωgμ̣,

where we have used the fact that all occuring unions and sums are
finite. In order to prove continuity, let f∈ S(μ;X). We estimate 

    ∫_Ωfμ̣_X    =∑_x∈ f[Ω]x·μ(A_f,x)_X
     ≤∑_x∈ f[Ω]x_Xμ(A_f,x)
     =∫_Ω∑_x∈ f[Ω]x_X_A_f,xμ̣
        =∫_Ωf(·)_Xμ̣=f_1.

The remaining assertions now follow from lem:simple_fcts_dense
by continuous extension (see cor:uniquecont).


The next proposition tells us how the Bochner-integral of a function behaves if we compose the function with a bounded or closed linear operator first.



Let f∈ L_1(μ;X), Y a Banach
space. 


  * Let B∈(X,Y). Then B∘ f∈ L_1(μ;Y) and 

    ∫_ΩB∘ fμ̣=B∫_Ωfμ̣.


  * If X_0⊆ X is a closed subspace and f(ω)∈ X_0
for μ-a.e. ω∈Ω, then ∫_Ωfμ̣∈ X_0.

  * (Theorem of Hille)Theorem of Hille Let A(A)⊆ X→ Y
be a closed linear operator and assume that f(ω)∈(A) for
μ-a.e. ω∈Ω and that A∘ f∈ L_1(μ;Y).
Then ∫_Ωfμ̣∈(A) and 

    A∫_Ωfμ̣=∫_ΩA∘ fμ̣.








  * Let (f_n)_n∈ be a sequence in S(μ;X) such that
f_n→ f μ-a.e. Then B∘ f_n∈ S(μ;Y) since

    (B∘ f_n)[Ω]=B[f_n[Ω]]

is finite and for y∈(B∘ f_n)[Ω]∖{0} we have
that 

    A_B∘ f_n,y=⋃_x∈ B^-1[{y}]∩ f_n[Ω]A_f_n,x∈Σ,

and thus, μ(A_B∘ f_n,y)<∞. Due to the continuity
of B we have that B∘ f_n→ B∘ f μ-a.e., hence
B∘ f is Bochner-measurable. Moreover, B∘ f(·)_Y≤Bf(·)_X,
which yields that B∘ f∈ L_1(μ;Y). By continuity of both B
and ∫_Ωμ̣, it suffices to check the interchanging property
for any f∈ S(μ;X) alone. However, this is clear, since for a simple function f

    B∘ f = B( ∑_x∈ X x ·_A_f,x)= ∑_x∈ X Bx ·_A_f,x,

where the sum is actually finite and hence, 

    ∫_ΩB∘ fμ̣    =∫_Ω∑_x∈ X Bx·_A_f,xμ̣= ∑_x∈ X∫_Ω Bx·_A_f,xμ̣
       =∑_x∈ XBx·μ(A_f,x)= B( ∑_x∈ X x·μ(A_f,x))=B ∫_Ωfμ̣,

where in the third equality we have used that Bx·_A_f,x is a simple function.

  * Let x'∈ X' with x'|_X_0=0. It follows from (a) that

    x'(∫_Ωfμ̣)=∫_Ωx'∘ fμ̣=0,

and since x' was arbitrary, it follows that ∫_Ωfμ̣∈ X_0
from the Theorem of Hahn–Banach.


  * Consider the space L_1(μ;X× Y). By assumption,
it follows that 

    (f,A∘ f)∈ L_1(μ;X× Y).

However, (f,A∘ f)(ω)=(f(ω),(A∘ f)(ω))∈ A⊆ X× Y
for μ-a.e. ω∈Ω, and since A is closed we can
use (b) to derive that 

    ∫_Ω(f,A∘ f)μ̣∈ A.

Let π_1,π_2 be the projection from X× Y to X
and Y, respectively. It then follows from part (a) that 

    π_1(∫_Ω(f,A∘ f)μ̣)=∫_Ωπ_1(f,A∘ f)μ̣=∫_Ωfμ̣,

and analogously for π_2. Using these equalities we derive from
eq:int in A that ∫_Ωfμ̣∈(A) and that
A∫_Ωfμ̣=∫_ΩA∘ fμ̣.



As a consequence of the latter proposition, we derive the fundamental
theorem of calculus for Banach space-valued functions.

fundamental theorem of calculusLet
a,b∈,a<b and consider the measure space (ab,ℬ(ab),λ),
where ℬ(ab) denotes the Borel-σ-algebra
of ab and λ is the Lebesgue measure. Let fab→ X
be continuously differentiable.[By this we mean that f is continuous on ab, continuously
differentiable on ab and f' has a continuous extension
to ab.] Then 

    f(b)-f(a)=∫_abf'λ̣.




Note first of all that continuous functions are Bochner-measurable (which can be easily seen using thm:Pettis). Thus, the integral
on the right-hand side is well-defined. Let φ∈ X'. Then φ∘ f[a,b]→
is continuously differentiable, and (φ∘ f)'(t)=(φ∘ f')(t).
Using prop:interchange_integral (a) together with the fundamental
theorem of calculus for the scalar-valued case we get 

    φ(∫_abf'λ̣)     =∫_ab(φ∘ f')λ̣=φ(f(b))-φ(f(a))
     =φ(f(b)-f(a)).

Since this holds for all φ∈ X', the assertion follows from
the Theorem of Hahn–Banach.

We conclude this section with a density result, which will be useful
throughout the course.

Let 𝒟⊆Ł(μ) be total
in Ł(μ) and H a Hilbert space. Then the set φ(·)xx∈ H, φ∈𝒟
is total in Ł(μ;H).


By lem:simple_fcts_dense, we know that S(μ;X) is dense in Ł(μ;H). Thus, it suffices to approximate _A x for some A∈Σ and x∈ H. For this, however, take a sequence (ϕ_n)_n in the linear hull of 𝒟 with ϕ_n→_A in Ł(μ) as n→∞. Then

    _Ax-ϕ_n x_Ł(μ;H) = x_H_A - ϕ_n_Ł(μ)→ 0   (n→∞).
Thus, the claim follows.


 Let 𝒟⊆Ł(μ) be total
in Ł(μ), H a Hilbert space, D_0⊆ H total in
H. Then φ(·)xx∈ D_0,φ∈𝒟
is total in Ł(μ;H).



The proof follows upon realising that the set φ(·)xx∈ D_0, φ∈𝒟
is total in the set φ(·)xx∈ H, φ∈𝒟. From here we just apply lem:dense sets.




§ THE TIME DERIVATIVE AS A NORMAL OPERATOR



Now let H be a Hilbert space over ∈{,}. For ν∈
and p∈1∞ we define the measure 

    μ_p,ν(A)∫_A^-pν tλ̣(t)

for A in the Borel-σ-algebra, ℬ(), of .
As our underlying Hilbert space for the time derivative we set 

    (;H)Ł(μ_2,ν;H).

In the same way we define 

    L_p,ν(;H) L_p(μ_p,ν;H)

for p∈1∞. If H= we abbreviate L_p,ν():=L_p,ν(;).

Our aim is to define the time derivative on (;H).
For this, we define the integral as an operator, which for ν≠0 turns out to be one-to-one and bounded. Then we introduce the time derivative as the inverse of this integral. The reason for doing it that way is to easily get a formula for the adjoint for the time derivative using the boundedness of the integral.

We start our considerations with the definition
of convolution operators in (;H). 

Let k∈ L_1,ν(). We define the convolution
operator

    k∗(;H)→(;H)

by 

    (k∗ f)(t)∫_k(s)f(t-s)ṣ,

which exists for a.e. t∈. Then, k∗ is linear and
bounded with k∗≤k_L_1,ν().



We first prove that s↦ k(s)f(t-s)∈ L_1(;H) for
a.e. t∈. The Bochner-measurability is clear since k and
f are both Bochner-measurable. Moreover, 

    ∫_(∫_k(s)f(t-s)_Hṣ)^2^-2ν tṭ
        =∫_(∫_k(s)^1/2^-ν/2sk(s)^1/2^-ν/2sf(t-s)_H^-ν(t-s)ṣ)^2ṭ
       ≤∫_(∫_k(s)^-ν sṣ)(∫_k(s)^-ν sf(t-s)_H^2^-2ν(t-s)ṣ)ṭ
        =k_L_1,ν()∫_k(s)∫_f(t-s)^2^-2ν(t-s)ṭ ^-ν sṣ
        =k_L_1,ν()^2f_(;H)^2,

which on the one hand proves that 

    ∫_k(s)f(t-s)_Hṣ<∞

for a.e. t∈ and on the other hand shows the norm estimate, once we have shown the Bochner-measurability of k∗ f. For proving the latter, we apply thm:Pettis. Since f is Bochner-measurable, we find a nullset N such that H_0  f[∖ N] is separable. Hence, for almost every t∈ we have 

    (k∗ f)(t)=∫_ k(s)f(t-s)ṣ=∫_∖ N k(t-s) f(s)ṣ∈ H_0

by prop:interchange_integral (b). Thus, k ∗ f is almost separably-valued. Moreover, for x' ∈ H' we have by prop:interchange_integral (a) 

    x'∘ (k∗ f)= k∗ (x'∘ f)

almost everywhere and thus, the weak Bochner-measurability follows from the fact that the convolution of two measurable scalar-valued functions is measurable. Since the linearity of k∗ is clear the proof is done.


For ν0 we define the operator 

    I_ν(;H)→(;H)

by 

    I_ν_0∞∗,     if ν>0,
    
    -_-∞0∗,     if ν<0.


Note that, by lem:convolution, I_ν is bounded with I_ν≤1/ν.



  For ν>0, f∈(,H) we have
  
    I_ν f(t) = _0∞∗ f (t) = ∫_0^∞ f(t-s)ṣ = ∫_-∞^t f(s)ṣ   (t∈).

  Analogously, for ν<0, f∈(,H) we have
  
    I_ν f(t) = -∫_t^∞ f(s)ṣ   (t∈).




Let ν0. Then I_ν is one-to-one
and (;H)C_c^1(R;H)@(;H),
the space of continuously differentiable, compactly supported functions
on  with values in H, is in the range of I_ν.



We just prove the assertion for the case when ν>0. Let f∈(;H)
satisfy I_νf=0. In particular, we obtain for all t∈∖ N
that 0=I_νf(t)=∫_-∞^tf(s)ṣ
for some Lebesgue null set, N⊆. Then for a,b∈∖ N
with a<b and x∈ H we have that 

    f^2ν(·)_ab· x_(;H)    =∫_f(t)^2ν t_ab(t)· x_H^-2ν tṭ
        =∫_a^bf(t)ṭx_H
        =(I_νf)(b)-(I_νf)(a)x_H=0.

Thus f=0. Indeed,  since ∖ N is dense in , ^2ν(·)_aba,b∈∖ N
is total in (). Hence, ^2ν(·)_ab· xa,b∈∖ N, x∈ H
is total in (;H) by lem:dense sets. This proves
the injectivity of I_ν. Moreover, if φ∈(;H)
then by cor:Hauptsatz we have 

    φ(t)=∫_-∞^tφ'(s)ṣ=(I_νφ')(t)  (t∈).*



For ν0 we define thetime derivative time derivative,
ν, on (;H) by 

    ν I_ν^-1.


Note that by lem:convolution and prop:int_nice,
ν is a closed linear operator for which (;H)⊆(ν).
Since 

    (;H)⊇φ· xφ∈(), x∈ H

we infer that ν is densely defined by lem:dense sets
and exer:C_cinfty dense. Moreover, since I_νφ'=φ
for φ∈(;H) we get that 

    νφ=φ';

that is, ν extends the classical derivative of continuously
differentiable functions. We shall discuss the actual domain of ν
in the next chapter. 

Let ν0. Then mathcalD_H@φ· xφ∈(), x∈ H
is a core for ν. Here, () denotes the
space of smooth functions on  with compact
support.



We first prove that 

    φ'φ∈()

is dense in (). As () is dense in ()
(see exer:C_cinfty dense), it suffices to approximate
functions in (). For this, let f∈().












We now define 

    φ_n(t)∫_-∞^tf(s) - f(s-n) ṣ   ν>0,
    ∫_-∞^tf(s) - f(s+n) ṣ   ν<0
                             (t∈,n∈).

Then φ_n∈() for each n∈ and 

    φ_n'(t)=
                     f(t)-f(t-n)    ν>0,
    
                     f(t)-f(t+n)    ν<0
                       (t∈,n∈).

Consequently, 

    φ_n'-f_()^2   = ∫_ |f(t-n)|^2  ^-2ν tṭ   ν>0,
    ∫_ |f(t+n)|^2 ^-2ν tṭ   ν<0
                                         
       = f_()^2 ^-2|ν|n→ 0   (n→∞),

which shows the density of eq:der_test_fct in (). By lem:dense sets we have that 

    φ'· xφ∈(), x∈ H

is total in (;H) and so ν[] is dense
in (;H). Now let f∈(ν) and ε>0.
By what we have shown above there exists some φ∈
such that 

    νφ-νf_(;H)≤ε.

Since ∂_t,ν^-1=I_ν is bounded with ∂_t,ν^-1≤1/ν, the latter implies
that

    φ-f_(;H)≤ε/ν,

and hence,  is indeed a core for ν. 



 For ν∈ the mapping

    exp(-ν):(;H)    →Ł(;H)
    
    f    ↦(t↦^-ν tf(t))

is unitary, and for ν,μ0 one has 

    exp(-ν)(ν-ν)exp(-ν)^-1=exp(-μ)(μ-μ)exp(-μ)^-1.




The proof is left as exer:derivative nu=00003D0.


By cor:nu=00003D0 we can now define 0. Let ν≠0. Then

    0exp(-ν)(ν-ν)exp(-ν)^-1.

Note that in view of cor:nu=00003D0, the assertion of prop:C_cinfty_core now also holds for ν=0.

Finally, we want to compute the adjoint of ν. 


Let ν∈. The adjoint of
ν is given by 

    ν^∗=-ν+2ν.

In particular, ν is a normal operator with ν1/2(ν+ν^∗)=ν, and 0 is skew-selfadjoint.



Let ν≠ 0 first.
Integrating by parts, one obtains 

    ∫_νφ(t)ψ(t)^-2ν tṭ    =∫_φ'(t)ψ(t)^-2ν tṭ
        =∫_φ(t)-ψ'(t)+2νψ(t)^-2ν tṭ

for φ,ψ∈(;H). Since (;H)
is a core for ν by prop:C_cinfty_core, the
latter shows 

    ν⊆-ν^∗+2ν.

Since we know that ν is onto, it suffices to prove that -ν^∗+2ν is one-to-one, since this would imply equality in the latter operator inclusion. For doing so, we apply thm:ran-kernerl to compute

    (-ν^∗ +2ν)=(-ν+2ν)^.

Moreover, we have that -ν+2ν is unitarily equivalent to --ν by cor:nu=00003D0 and since -ν is onto, so is -ν+2ν and thus (-ν^∗ +2ν)=(;H)^={0}, which yields the assertion.

The case ν=0 follows directly from the definition of 0.





§ COMMENTS


Standard references for Bochner integration and related results are <cit.>.

Considering the derivative operator in an exponentially weighted space
goes back (at least) to Morgenstern <cit.>, where
ordinary differential equations were considered in a classical
setting. In fact, we shall return to this observation in the next
chapter when we devote our study to some implications of the already developed
concepts on ordinary and delay differential equations. 

A first occurence of the derivative operator in exponentially weighted
L^2-spaces can be found in <cit.>, where a corresponding
spectral theorem has been focussed on. We will prove in a later chapter
that the spectral representation of the time-derivative as a multiplication operator can
be realised by a shifted variant of the Fourier transformation –
the so-called Fourier–Laplace transformation. 

In an applied context, the time derivative operator discussed here
has been introduced in <cit.>. 



§ EXERCISES

tocsectionExercises


A sequence (φ_n)_n in (^d)
is called a δ-sequence,delta-sequence,@δ-sequence
if 




  * φ_n≥0 for n∈,


  * φ_n⊆-1/n1/n^d
for n∈,


  * ∫_^dφ_n=1 for n∈.



Let φ∈(^d) with φ⊆-11^d,
φ≥0 and ∫_^dφ=1. Prove that (φ_n)_n
given by φ_n(x) n^dφ(nx) for x∈^d, n∈
defines a δ-sequence. Moreover, give an example for such a
function φ. 



 It is well-known
that _II-dimensional bounded interval
is total in Ł(^d).



  * Let φ∈(^d), f∈Ł(^d). Define as usual f∗φ(x↦∫_^d f(x-y) φ(y) ỵ).
Prove that f∗φ∈ C^∞(^d) with ∂^α(f∗φ)=f∗∂^αφ
for all α∈_0^d, where ∂^αφ=∂_1^α_1⋯∂_d^α_dφ.
Moreover, prove that f∗φ⊆ f+φ.


  * Let (φ_n)_n be a δ-sequence and f∈Ł(^d).
Show that f∗φ_n→ f in Ł(^d) as n→∞.

Hint: Prove that _I∗φ_n→_I in Ł(^d)
for all d-dimensional intervals and use that f∗φ_n_2≤f_2
(see also lem:convolution).


  * Prove that (^d) is dense in Ł(^d).





Let a<b, X_0,X_1,X_2 be Banach
spaces, fab→ X_0 and gab→ X_1 both continuously differentiable,
ℓ X_0× X_1→ X_2 bilinear and continuous. Prove
that hab→ X_2 given by

    h(t)ℓ(f(t),g(t))  (t∈ab)

is continuously differentiable with 

    h'(t)=ℓ(f'(t),g(t))+ℓ(f(t),g'(t))  (t∈ab).

If f,f',g,g' have continuous extensions to ab, prove
the integration by parts formula: 

    ∫_a^bℓ(f'(t),g(t))ṭ=ℓ(f(b),g(b))-ℓ(f(a),g(a))-∫_a^bℓ(f(t),g'(t))ṭ.




For ν0, show that I_ν=1/ν.



 Prove cor:nu=00003D0.



 Let ν∈ and H be a complex Hilbert
space. Prove that σ(ν)⊆ı t+νt∈,
where 0 is defined in cor:adjoint_time_derivative.

Hint: For f∈(ν), z∈ compute (z-ν)ff_(;H)
by using cor:adjoint_time_derivative. For proving the
surjectivity of z-ν for a suitable z, use the formula 

    (z-ν)=(z^∗-ν^∗)^.


Remark: Later we will see that, actually, σ(ν)=ı t+νt∈.



Consider the differential equation

    (ν^2-1)u=_[-1,1].

Since ν^2-1=(ν-1)(ν+1),
it follows by exer:spectrum_time_Der that there is a
unique u∈() solving this equation if ν∉{-1,1}.
Compute these solutions.

Hint: For u∈(ν) use the fact that u is necessarily continuous (which we shall establish in the next lecture).


[heading=subbibliography]



CHAPTER: ORDINARY DIFFERENTIAL EQUATIONS


In this lecture, we discuss a first application of the time derivative
operator constructed in the previous lecture. More precisely, we analyse
well-posedness of ordinary differential equations and will at the
same time provide a Hilbert space proof of the classical Picard–Lindelf
theorem. We shall furthermore see that the abstract theory developed
here also allows for more general differential equations to be considered.
In particular, we will have a look at so-called delay differential equations
with finite or infinite delay; neutral differential equations are
considered in the exercises section.

We start with some information on the time derivative and its
domain.



§ THE DOMAIN OF THE TIME DERIVATIVE AND THE SOBOLEV EMBEDDING THEOREM


Let H be a Hilbert space.
Readers familiar with the notion of Sobolev spaces might
have already realised that the domain of ν can be described as (;H)-functions
with distributional derivative lying in (;H). In order to
stress this, we include the following result. Later on, we have the
opportunity to have a more detailed look at Sobolev spaces in more general
contexts.

Let ν∈ and f,g∈(;H).
Then the following conditions are equivalent: 



  * f∈(ν) and νf = g.

  * For all ϕ∈()
we have 

    -∫_ϕ'f=∫_ϕ g.





Assume that f∈(ν). By prop:C_cinfty_core
and cor:adjoint_time_derivative, we have that =φ· xφ∈(), x∈ H⊆(ν^*) (which also holds for ν=0)
and 

    νfψ· x_=f(-ψ'+2νψ)· x_

for all x∈ H and ψ∈(). Hence, we obtain for all
ψ∈() 

    ∫_(-ψ'+2νψ)f^-2ν·=∫_ψνf^-2ν·;

putting ϕ^-2ν·ψ and using that multiplication
by ^-2ν· is a bijection on (), we deduce the
claimed formula with g=νf.

On the other hand, the equation involving g applied to ϕ=^-2ν·ψ
for ψ∈() implies that

    ∫_(-ψ'+2νψ)f^-2ν·=∫_ψ g^-2ν·.

Testing this equation with x∈ H 
yields

    gψ· x_=f(-ψ'+2νψ)· x_=f(-νψ· x +2νψ· x)_.

Since  is dense in (ν) by prop:C_cinfty_core, we infer that

    gh_=f(-νh+2ν h)_

for all h∈(ν). Now, cor:adjoint_time_derivative,
yields

    gh_=fν^*h_  (h∈(ν^*)).

Thus, f∈(ν^**)=(ν) and νf=g. 


The next result confirms that functions in the domain of ν
are continuous. This result was announced in exer:inverse-depends-on-nu and  is known as the Sobolev embedding theorem. Here,
we make use of the explicit form of the domain of ν as being
the range space of the integral operator I_ν. We defineC_nu(R;H)@C_ν(;H)

    C_ν(;H)f→ Hf continuous, f_ν,∞sup_t∈^-ν tf(t)_H<∞
and regard it as being
endowed with the obvious norm. 

 Let ν∈. Then every f∈(ν) has a continuous
representative, and the mapping 

    (ν)∋ f↦ f∈ C_ν(;H)

is continuous. 



We restrict ourselves to the case when ν>0; the remaining cases can be proved by invoking cor:nu=00003D0.
Let f∈(ν). By definition, we find g∈(;H)
such that f=ν^-1g=I_νg. Then for all t∈ we compute

    ∫_-∞^tg(τ)τ̣   
     =∫_-∞^tg(τ)^-ντ^νττ̣≤√(∫_-∞^tg(τ)^2^-2νττ̣)√(∫_-∞^t^2νττ̣)
       ≤νf_√(1/2ν)^ν t.

Thus, g is integrable on -∞t for all t∈ and dominated convergence implies that

    f = ( t↦∫_-∞^t g(s)ṣ)

is continuous.
Moreover, for t∈ we obtain

    f(t)≤∫_-∞^tg(τ)τ̣≤νf_√(1/2ν)^ν t

which yields the claimed continuity.



For all f∈(ν), we have that ^-ν tf(t)_H→0
as t→±∞. 


The proof is left as exer:vanish_at_inf.







§ THE PICARD–LINDELF THEOREM


The prototype of the Picard–Lindelf theorem will be formulated
for so-called uniformly Lipschitz continuous functions.

We first need a preparation.

  Let X be a Banach space. Then we define 
  
    S_(;X) f→ Xf simple,  f compact

  to be the set of simple functions from  to X with compact support.




  Let X be a Banach space and ν,η∈. Then S_(;X) is dense in (;X)∩η(,X); that is, for all f∈(;X)∩η(;X) there exists (f_n)_n in S_(;X) such that f_n→ f in both (;X) and η(;X). 
	 In particular, S_(,X) is dense in (;X).



  Let f∈(;X)∩η(;X).
  Then for all n∈ we have that _-nnf∈(;X)∩η(;X)
  and _-nnf→ f in (;X) and in η(;X)
  as n→∞. 
  For n∈ let (f̃_n,k)_k be in
  S(μ_2,ν;X) such that f̃_n,k→_-nnf
  in (;X) as k→∞. We
  put f_n,k_-nnf̃_n,k∈ S_(;X). Then
  f_n,k→_-nnf in (;X) and in η(;X)
  as k→∞.


In order to define the notion of uniformly Lipschitz continuous functions, we first need the Lipschitz semi-norm.

  Let X_0, X_1 be normed spaces, and F X_0→ X_1 Lipschitz continuous. Then 
  
    F_sup_x,y∈ X_0 
     x≠ yF(x)-F(y)/x-y

is the Lipschitz semi-norm of F. 



Let H_0,H_1 be Hilbert spaces, μ∈. Then a function F S_(;H_0)→⋂_ν≥μ(;H_1)
is called uniformly Lipschitz continuous, if for all ν≥μ
we have that F considered in (;H_0)×(;H_1)
is Lipschitz continuous, and for the unique Lipschitz continuous extensions
F^ν, ν≥μ, we have that

    sup_ν≥μF^ν_<∞.
 



  Another way to introduce uniformly Lipschitz continuous mappings is the following.
  Let H_0,H_1 be Hilbert spaces, μ∈. Let (F^ν)_ν≥μ be a family of Lipschitz continuous mappings F^ν(;H_0)→(;H_1) such that
  
    sup_ν≥μF^ν_<∞

  and the mappings are consistent in the sense that for all ν,η≥μ and f∈(;H_0)∩η(;H_0) we have
  
    F^ν(f) = F^η(f).

  Then, for ν≥μ and f∈ S_(;H_0) we have F^ν(f)∈⋂_η≥μη(;H_1) and F^ν|_S_(;H_0) is uniformly Lipschitz continuous.



 Let H be a Hilbert space, μ∈ and F S_(;H)→⋂_ν≥μ(;H)
uniformly Lipschitz continuous, with Lsup_ν≥μF^ν_.
Then for all ν>max{L,μ} the equation 

    νu_ν=F^ν(u_ν)

admits a unique solution u_ν∈(ν). Furthermore,
for all ν>max{L,μ} the following properties hold: 


  * 
If F^ν(u_ν) is continuous in a neighborhood of a∈,
then u_ν is continuously differentiable in a neighborhood of a. 

  * 
For all a∈, _-∞au_ν is the unique fixed
point v∈(;H) of _-∞aν^-1F^ν, that is, v uniquely solves

    v=_-∞aν^-1F^ν(v).


  * 
For all η≥ν we have that u_ν=u_η. 

  * 
For all f∈(;H) the equation 

    νv=F^ν(v)+f

admits a unique solution v_ν,f∈(ν), and if f,g∈(;H)
satisfy f=g on -∞a for some a∈ℝ,
then v_ν,f=v_ν,g on -∞a. 




Define Φ(;H)→(;H) by 

    Φ(u)=ν^-1F^ν(u).

Since ν^-1≤1/ν and ν>L
it follows that Φ is a
contraction and thus admits a unique fixed point, which by definition
solves the equation in question. Moreover, we have that u_ν=Φ(u_ν)=ν^-1F^ν(u_ν)∈(ν).

Differentiability of u_ν as in <ref> follows from exer:weak_diff+cont and the continuity of F^ν(u_ν).

For the unique existence asserted in <ref>, note that the unique existence of v_ν,f follows from the above considerations after
realising that Φ(v)ν^-1F^ν(v)+ν^-1f
defines a contraction in (;H). For the remaining statements in <ref> and the statements in <ref> and <ref>,
we need some prerequisites. 



  Let H_0, H_1 be Hilbert spaces, and F(;H_0)→(;H_1) Lipschitz continuous. Then, F is called causalcausal if for all a∈ and all f,g∈(;H_0) with
  f=g on -∞a, we have that F(f)=F(g) on -∞a.



In the following, we will frequently  make use of the following easy estimates: 
Let ν∈, a∈. If f∈(;H) with f⊆-∞a then f∈⋂_η≤νη(;H) and

    f_η(;H)≤^(ν-η)af_(;H)   (η≤ν).

Likewise, if f⊆a∞, we get f∈⋂_ρ≥νρ(;H) with

    f_ρ(;H)≤^(ν-ρ)af_(;H)   (ρ≥ν).




Let H_0,H_1 be Hilbert spaces, μ∈, F S_(;H_0)→⋂_ν≥μ(;H_1)
uniformly Lipschitz continuous. Then the following statements hold: 


  *  F^ν is causal for all ν≥μ. 

  *  The mapping ν^-1F^ν is causal if ν≥max{μ,0} and ν≠ 0. 

  *  For all ν≥η≥μ, we have that F^ν=F^η on (;H_0)∩μ(;H_0). 



 
<ref> 
We divide the proof into three steps.

(i) Let ν≥μ. In order to show causality of F^ν, we first note that it suffices to have F^ν(f) = F^ν(g) on -∞a for all f,g∈ S_(;H_0) with f=g on -∞a. Indeed, let f,g∈(;H) with f=g on -∞a for some a∈. By lem:simple_fcts_compact_spt we find
(f_n)_n and (g̃_n)_n in S_(;H_0) such that
f_n→ f and g̃_n→ g in (;H_0). Next,
_-∞af_n→_-∞af=_-∞ag
as n→∞ in (;H_0). Thus, putting g_n_-∞af_n+_a∞g̃_n
for all n∈ we obtain that g_n→ g in (;H_0).
Since F^ν(f_n) = F^ν(g_n) on -∞a for all n∈ and F^ν(;H_0)→(;H_1) is continuous, taking the limit n→∞ yields F^ν(f) = F^ν(g) on -∞a.

(ii) Let a∈, c≥0 and f∈ S_(;H_0) such that f=0 on -∞a, g∈⋂_ν≥μ(;H_1) such that
g_(;H_1)≤ cf_(;H_0) for all ν≥μ. Then

    ∫_-∞^a g(t)_H_1^2 ^2ν(a-t)ṭ≤∫_g(t)_H_1^2 ^2ν(a-t)ṭ≤ c^2 ∫_a^∞f(t)_H_0^2 ^2ν(a-t)ṭ→ 0

as ν→∞. Since ^2ν(a-t)→∞ as ν→∞ for all t<a, the monotone convergence theorem implies g=0 on -∞a.

(iii) Let f,g∈ S_(;H_0) such that f=g on -∞a for some a∈. Then f-g=0 on -∞a. Since F is uniformly Lipschitz continuous, with Lsup_ν≥μF^ν_ we obtain F^ν(f)-F^ν(g)_(;H_1)≤ Lf-g_(;H_0) for all ν≥μ. 
By (ii) we conclude F^ν(f)=F^ν(g) on -∞a for all ν≥μ, which by (i) yields the assertion.

The statement in <ref> directly follows from <ref>. 
Note that ν^-1F^ν is uniformly Lipschitz continuous only for ν>0.

Let us prove <ref>.
Since F^ν(f)=F(f)=F^η(f) for f∈ S_(;H_0), the set S_(;H_0) is dense in (;H_0)∩μ(;H_0) by lem:simple_fcts_compact_spt, and F^ν and F^η are Lipschitz-continuous, we obtain the assertion.


 The remaining part in <ref>:
Let f,g∈(;H) with f=g on -∞a. Since ν>L≥ 0,
we compute using lem:indepCaus<ref> and causality of ν^-1 that

    _-∞av_ν,f    =_-∞aν^-1F^ν(v_ν,f)+_-∞aν^-1f
        =_-∞aν^-1F^ν(_-∞av_ν,f)+_-∞aν^-1_-∞af
        =_-∞aν^-1F^ν(_-∞av_ν,f)+_-∞aν^-1_-∞ag.

The same computation also yields that

    _-∞av_ν,g=_-∞aν^-1F^ν(_-∞av_ν,g)+_-∞aν^-1_-∞ag.

It is easy to see that u↦_-∞aν^-1F^ν(u)+_-∞aν^-1_-∞ag
defines a contraction in (;H). Hence, the contraction mapping
principle implies that _-∞av_ν,f=_-∞av_ν,g.

The  statement in <ref> follows from the fact that u↦_-∞aν^-1F^ν(u)
defines a contraction and lem:indepCaus<ref>.

For the proof of <ref>, we observe that for all n∈, we have _-∞nu_η∈(;H)∩η(;H).
Hence, by <ref> and lem:indepCaus<ref>, it follows that

    _-∞nu_η    =_-∞nη^-1F^η(_-∞nu_η)
    =_-∞nν^-1F^ν(_-∞nu_η).

As _-∞nu_ν satisfies the same fixed point
equation, we deduce _-∞nu_η=_-∞nu_ν
for all n∈, which yields the assertion. 

As a first application of thm:PicLind1 we state and prove
the classical version of the Theorem of Picard–Lindelf. 


  Let H be a Hilbert space, Ω⊆× H be open, fΩ→ H continuous, (t_0,x_0)∈Ω. 
Assume there exists L≥0
such that for all (t,x),(t,y)∈Ω we have 

    f(t,x)-f(t,y)≤ Lx-y.

Then, there exists δ>0 such that the initial value problem

    u'(t)=f(t,u(t))     (t∈t_0t_0+δ),
    
    u(t_0)=x_0,

admits a unique continuously differentiable solution, ut_0t_0+δ→ H, which satisfies
(t,u(t))∈Ω for all t∈t_0t_0+δ. 



First of all we observe that we may assume, without loss of generality, that
x_0=0. Indeed, to solve the initial value problem 

    v'(t)=f(t,v(t)+x_0)     (t∈t_0t_0+δ),
    
    v(t_0)=0,

for a continuously differentiable vt_0t_0+δ→ H
is equivalent to solving the problem in thm:PicLind_class
for u by setting u=v+_t_0t_0+δx_0. Appropriately
shifting the time coordinate, we may also assume that t_0=0.

Thus, let (0,0)∈Ω. Let also [0,δ']×0ε⊆Ω
for some δ',ε>0. Denote by P H→ H
the orthogonal projection onto 0ε.
By exer:ProjC, P is Lipschitz continuous with Lipschitz
semi-norm bounded by 1. We then define 

    F S_(; H)    →⋂_ν≥0(; H)
    
    g    ↦(t↦_0δ'(t)f(t,P(g(t))))

and will prove that F is well-defined and uniformly Lipschitz continuous. Since the mapping t↦_0δ'(t)f(t,0)
is supported on 0δ', we obtain for ν≥0 that F(0)∈(;H).
Moreover, for ν≥0 and g,h∈ S_(; H) we estimate

    F(g)-F(h)_(;H)^2
        =∫_F(g)(t)-F(h)(t)^2^-2ν tṭ
     =∫_0^δ'f(t,P(g(t)))-f(t,P(h(t)))^2^-2ν tṭ
       ≤ L^2∫_0^δ'P(g(t))-P(h(t))^2^-2ν tṭ≤ L^2∫_0^δ'g(t)-h(t)^2^-2ν tṭ
       ≤ L^2g-h_(;H)^2,

which shows that F is well-defined and uniformly Lipschitz continuous.

By thm:PicLind1, there exists v∈(ν) with
ν>L such that 

    νv=F^ν(v).

We read off from v=ν^-1F^ν(v) that v=0 on -∞0, and that v is continuous by thm:Sobolev_emb. 
Moreover, we obtain that

    v(t)=∫_-∞^t_0δ'(τ)f(τ,P(v(τ)))τ̣=∫_0^min{t,δ'}f(τ,P(v(τ)))τ̣,

from which we read off that v is continuously differentiable on
0δ' since f and P are also continuous. The
same equality implies for 0<t≤δmin{ε/M,δ'},
where Msup_(t,x)∈0δ'×0εf(t,x),
that 

    v(t)≤∫_0^tf(τ,P(v(τ)))τ̣≤ Mδ≤ε.

Thus, (t,v(t))∈0δ'×0ε⊆Ω
for all 0≤ t≤δ and so Pv(t)=v(t) for 0≤ t≤δ.
Thus, u v|_0δ satisfies eq:IVP_ODE.

Finally, concerning uniqueness, let ũ0δ→ H
be a continuously differentiable solution of eq:IVP_ODE.
Let ṽ be the extension of ũ by 0 to the whole
of . Then we get that 

    _-∞δṽ    =_-∞δ∫_0^·_0δ'(τ) f(τ,ṽ(τ))τ̣
        =_-∞δ∫_-∞^·_0δ'(τ)f(τ,P(ṽ(τ)))τ̣
        =_-∞δν^-1F^ν(_-∞δṽ).

Since _-∞δv is the unique solution of the equation
w=_-∞δν^-1F^ν(w), we obtain that _-∞δṽ=_-∞δv,
which yields u=ũ. 


 The reason for the proof of the classical Picard–Lindelöf theorem being seemingly complicated is two-fold. First of all, the Hilbert space solution theory is for Ł-functions rather than continuous (or continuously differentiable) functions. The second, maybe more important point is that the Hilbert space Picard–Lindelöf asserts a solution theory, which provides global existence in the time variable. The main body of the proof of the classical Picard–Lindelöf theorem presented here is therefore devoted to `localisation' of the abstract theorem. Furthermore, note that the method of proof for obtaining uniqueness and the admittance of the initial value rests on causality. This effect will resurface when we discuss partial differential equations.




§ DELAY DIFFERENTIAL EQUATIONS


In this section, our study will not be as in depth as done for the local Picard–Lindelf theorem. 
Of course, the solution theory afforded would not be a very good one, if it was only applicable to, arguably,  the easiest case of ordinary differential equations.
We shall see next that the developed theory  applies
to more elaborate examples.

In what follows, let H be a Hilbert space over . We start out with a delay differential equation with  so-called `discrete delay'.
For this, we introduce, for h∈, the time-shift operator

    τ_h S_(;H)    →⋂_ν∈(;H),
    
    f    ↦ f(·+h).




Let μ∈. The mapping τ_h S_(;H) →⋂_ν≥μ(;H) is uniformly Lipschitz continuous
if and only if h≤ 0.
Moreover, for ν∈ we have 

    τ_h_(;H)=^hν.




Let f∈ S_(;H). Then for ν∈ we compute

    τ_hf_(;H)^2    =∫_f(t+h)^2^-2ν tṭ
     =∫_f(t)^2^-2ν(t-h)ṭ
        =f_(;H)^2^2ν h.

Since sup_ν≥ν^2ν h <∞ if and only if h≤ 0 we obtain the equivalence. Moreover, the above equality also also yields the norm of τ_h in (;H).

We will reuse τ_h for the Lipschitz continuous extensions
to (;H). The well-posedness theorem for delay equations with discrete delay
is contained in the next theorem. We note here that we only formulate the respective
result for right-hand sides that are globally Lipschitz continuous.
With a localisation technique, as has already been carried out for the classical
Picard–Lindelf theorem, it is also possible to obtain local results. 

 Let H be a Hilbert
space, μ∈, N∈, h_1,…,h_N∈-∞0, and

    G S_(;H^N)→⋂_ν≥μ(;H)

uniformly Lipschitz. Then there exists an η∈ such that for
all ν≥η the equation 

    νu=G^ν(τ_h_1u,…,τ_h_Nu)

admits a solution u∈(ν) which is unique in ⋃_ν≥η(;H).
Moreover, for all a∈ the function u_a_-∞au
satisfies 

    u_a=_-∞aν^-1G^ν(τ_h_1u_a,…,τ_h_Nu_a).




The assertion follows from thm:PicLind1 applied to F G∘(τ_h_1,…,τ_h_N). 

Next, we formulate an initial value problem for a subclass of the
latter type of equations. 

 
Let h>0, f0× H× H→ H
continuous, and f(·,0,0)∈μ(; H) for some μ>0.
Assume that there exists L≥0 with 

    f(t,x,y)-f(t,u,v)≤ L(x,y)-(u,v)  ((t,x,y),(t,u,v)∈0× H× H).

Let u_0∈ C([-h,0]; H). Then there exists η∈
such that for all ν≥η the initial value problem 

    u'(t)=f(t,u(t),u(t-h))     (t>0),
    
    u(τ)=u_0(τ)     (τ∈-h0)

admits a unique continuous solution u-h∞→ H,
continuously differentiable on 0∞. 



For t<0 let f(t,·,·) 0.
We define F S_(; H)→⋂_ν≥μ(; H)
by 

    F(ϕ)(t)
        f(t,ϕ(t)+_0∞(t)u_0(0),ϕ(t-h)+_0∞(t-h)u_0(0)+_0h(t)u_0(t-h))

for all t∈. It is easy to see that F is uniformly Lipschitz continuous.
Thus, by thm:PicLind1, we find η≥μ such that
for all ν≥η the equation 

    νv=F^ν(v)

admits a solution v∈⋂_ν≥η(ν) which is unique
in ⋃_ν≥η(; H). Note that F^ν(v)⊆0∞.
Hence, v=0 on -∞0. By thm:Sobolev_emb,
we obtain that v(0)=0. We claim that u v+_0∞(·)u_0(0)+_-h0u_0
is a solution of eq:dde. First of all note that u
is continuous on -h∞. Next, for h> t>0 we have that t-h< 0
and thus v(t-h)=0 and so we see that

    F^ν(v)(t)     =f(t,v(t)+_0∞(t)u_0(0),v(t-h)+_0∞(t-h)u_0(0)+_0h(t)u_0(t-h))
        =f(t,u(t),u_0(t-h)).

Similarily, for t≥ h we obtain

    F^ν(v)(t)  =f(t,u(t),u(t-h))

and thus, by continuity of f, u_0 and u, it follows that v is
continuously differentiable on 0∞ and 

    u'(t)=v'(t)=νv(t)=f(t,u(t),u(t-h)).

It remains to show uniqueness. For this, let w-h∞→ H
be a solution of eq:dde. Then

    w(t)=u_0(0)+∫_0^t f(s,w(s),w(s-h))ṣ   (t≥ 0)

and w(t)=u_0(t) if t∈-h0. We set  ṽ w-_0∞(·)u_0(0)-_-h0u_0 and infer

    ṽ(t)   = ∫_0^t f(s,w(s),w(s-h))ṣ
       =∫_-∞^t f(s,ṽ(s)+_0∞(s)u_0(0),
                           ṽ(s-h)+_0∞(s-h)u_0(0)+_0h(s)u_0(s-h))ṣ

for all t∈. For  a∈ we set ṽ_a_-∞aṽ∈⋂_ν∈(; H)
and obtain, using the above formula for ṽ,

    ṽ_a=_-∞aν^-1 F^ν(ṽ_a).

 By uniqueness of the solution of 

    _-∞av=_-∞aν^-1F^ν(_-∞av)

it follows that ṽ_a=_-∞av for all a∈
and, thus, u=w. 

The equation to come involves the whole history of the unknown; that is, the unknown evaluated at -∞0.
For a mapping u→ H we define the mapping 

    u_(·)∋ t↦(0∋θ↦ u(t+θ)∈ H).



Let μ>0. Then 

    Θ S_(;H)    →⋂_ν≥μ(;Ł(0;H))
    
    u    ↦ u_(·)

is uniformly Lipschitz continuous. More precisely, for all ν>0 we have

    Θ^ν=1/√(2ν).




Let u∈ S_(;H). Then we compute 

    Θ u_(;Ł(0;H))^2    =∫_∫_0u(t+θ)^2θ̣^-2ν tṭ
     =∫_∫_0u(t)^2^-2ν(t-θ)θ̣ṭ
        =1/2ν∫_u(t)^2^-2ν tṭ.*



 Let H be a Hilbert space, μ∈ and let
Φ S(;Ł(0;H))→⋂_ν≥μ(;H)
be uniformly Lipschitz. Then, there exists η>0 such that for all
ν≥η the equation 

    νu=Φ^ν(u_(·))

admits a solution u∈⋂_ν≥η(ν) unique in
⋃_ν≥η(;H). 



This is another application of thm:PicLind1. 




§ COMMENTS


In a way, the proof of thm:PicLind_class is standard
PDE-theory in a nutshell; a solution theory for L_p-spaces is
used to deduce existence and uniqueness of solutions and a posteriori
regularity theory provides more information on the properties of the
solution.

Note that – of course – other proofs are available for the Picard–Lindelf
theorem. We chose, however, to present this proof here in order to
provide a perspective on classical results. Furthermore, we mention
that in order to obtain unique existence for the solution, it suffices
to assume that f satisfies a uniform Lipschitz condition with respect
to the second variable and that f is measurable. Continuity
of f is needed in order to obtain C^1-solutions.

A more detailed exposition and more examples of the theory applied
to delay differential equations can be found in <cit.>
and – in a Banach space setting – <cit.>.

There is also a way of dealing with delay differential equations by
expanding the state space the problem is formulated in. In this case,
it is possible to make use of the rich theory of C_0-semigroups.
We refer to <cit.> for this.

Causality is one of the main concepts for evolutionary equations. We have provided this notion for mappings defined on -type spaces only. The situation becomes different, if one considers merely densely defined mappings. Then it is a priori unclear, whether for a Lipschitz continuous mapping the continuous extension is also causal. For this we refer to exer:causal_cont below and to <cit.>, and <cit.> as well as to references mentioned there.



§ EXERCISES

tocsectionExercises


(a) Let X be a Banach space, uab→ X
continuous. Show that vab→ X given by 

    v(t)=∫_a^tu(τ)τ̣

is continuously differentiable with v'(t)=u(t).

(b) Let H be a Hilbert space, and ν∈. Let u∈(ν)
with νu continuous. Show that u is continuously differentiable
and u'=νu. 



Prove cor:vanish_at_inf. 



Let H be a Hilbert space.
Show that 

    (ν)↪ C_ν^1/2(;H)f∈ C_ν(;H)^-ν·f is -Hlder continuous,

where a function g→ H is said to be 1/2-Hlder
continuous, if 

    sup_s,t∈
     t≠ sg(t)-g(s)/t-s^1/2<∞.




Let H be a Hilbert space, C⊆ H closed
and convex. Show that the orthogonal projection, P, of H onto C
defines a Lipschitz continuous mapping with Lipschitz semi-norm bounded
by 1. 



Let h×0×^n→^n be continuous satisfying

    h(t,s,x)-h(t,s,y)≤ Lx-y

with h(·,·,0)=0. Let R>0 and u_0∈ C(0;^n) have
compact support. Show that the initial value problem 

    u'(t)=∫_-R^0h(t,s,u_(t)(s))ṣ    (t>0),
    
    u(t)=u_0(t)     (t≤0)

admits a unique continuous solution u→^n, which is
continuously differentiable on 0.

Hint: Modify Θ from lem:history. 



Let H be a Hilbert space.
Show that for a uniformly Lipschitz continuous Φ S(;Ł(0;H)^2)→⋂_ν≥μ(;H)
the equation 

    νu=Φ^ν(u_(·),(νu)_(·))

admits a unique solution u∈(ν) for ν large enough. 



Let D⊆Ł() be dense and suppose that F D⊆Ł()→Ł()
admits a Lipschitz continuous extension F^0.



  * Show that F^0 is causal if and only if for all ϕ∈ S_(;) and all a∈ there exists L≥ 0 such that

    _-∞a·(F(f)-F(g))ϕ_Ł()≤ L _-∞a·(f-g)_L_2()

for all f,g∈ D; that is,
the mapping 

    (D,_-∞a·(·-·)_Ł())∋ f↦ F(f) ∈(Ł(),_-∞a·(·-·)ϕ_Ł())

is Lipschitz continuous.

  * For a∈ let (F)∩(F_-∞a)
be dense in Ł() and if f,g∈ D=(F) and f=g on -∞a then also F(f) = F(g) on -∞a. Show that F^0 is causal.

  * Assume that for all f,g∈ D that f=g on -∞a
implies that F(f)=F(g) on -∞a. Show that this is not sufficient for F^0 to be causal. 
Hint: Find a dense subspace D=(F) so that the first condition in (b) is not satisfied.




[heading=subbibliography]



CHAPTER: THE FOURIER–LAPLACE TRANSFORMATION AND MATERIAL LAW OPERATORS


In this chapter we introduce the Fourier–Laplace transformation
and use it to define operator-valued functions of ν; the
so-called material law operators. These operators will play a crucial
role when we deal with partial differential equations. In the equations
of classical mathematical physics, like the heat equation, wave equation
or Maxwell's equation, the involved material parameters, such as heat
conductivity or permeability of the underlying medium, are incorporated
within these operators and hence the name “material law”. We start
our chapter by defining the Fourier transformation and proving Plancherel's
theorem in the Hilbert space-valued case, which states that the Fourier
transformation defines a unitary operator on Ł(;H).

Throughout, let H be a complex Hilbert space.




§ THE FOURIER TRANSFORMATION


We start by defining the Fourier transformation on L_1(;H). 


For f∈ L_1(;H) we define the Fourier transformFourier transform f̂ of f by 

    f̂(s)1/√(2π)∫_^-ı stf(t)ṭ  (s∈).



We also introduce 

    (;H)Cb(R;H)@(;H) f→ Hf continuous, bounded
 endowed with the sup-norm, ·_∞.


 Let f∈ L_1(;H). Then f̂∈(;H) and lim_t→∞f̂(t) = 0.
Moreover, 

    f̂ _∞≤1/√(2π)f_1.




First, note that f̂ is continuous by dominated convergence
and bounded with 

    f̂ _∞≤1/√(2π)f_1.

This shows that the mapping 

    L_1(;H)    →(;H),   f ↦f̂

defines a bounded linear operator. Moreover, for φ∈(;H) we compute

    φ̂(s)     =1/√(2π)∫_^-ı stφ(t)ṭ
     =1/√(2π)1/ı s∫_^-ı stφ'(t)ṭ

for s0 and thus, 

    lim sup_s→∞φ̂(s)≤lim sup_s→∞1/s1/√(2π)φ'_1=0,

which shows that lim_s→∞φ̂(s)=0. By the facts that
(;H) is dense in L_1(;H) (see lem:dense sets),
f∈(;H)lim_t→∞f(t)=0 is a closed subspace of (;H) and the operator
in eq:fourier_on_L1 is bounded, the assertion follows. 

It is our main goal to extend the definition of the Fourier transformation
to functions in Ł(;H). For doing so, we make use of the Schwartz space
of rapidly decreasing functions. 

We define 

    𝒮(;H)f∈ C^∞(;H)∀ n,k∈: (t↦ t^kf^(n)(t))∈(;H)

to be the Schwartz spaceSchwartz space of rapidly decreasing functions on  with values
in H. 

As usual we abbreviate 𝒮()𝒮(;).

𝒮(;H) is a Frchet space with respect to the seminorms

    𝒮(;H)∋ f↦sup_t∈t^kf^(n)(t)  (k,n∈).

Moreover, 𝒮(;H)⊆⋂_p∈1∞L_p(;H).
Indeed, 𝒮(;H)⊆ L_∞(;H) by definition,
and for 1≤ p<∞ we have that 

    ∫_f(t)^pṭ    =∫_1/(1+t)^2p(1+t)^2f(t)^pṭ
       ≤sup_t∈(1+t)^2f(t)^p ∫_1/(1+t)^2pṭ<∞.




 For f∈𝒮(;H)
we have f̂∈𝒮(;H) and the mapping 

    𝒮(;H)→𝒮(;H),   f↦f̂

is bijective. Moreover, for f,g∈ L_1(;H) we have that 

    ∫_f̂(t)g(t)ṭ=∫_f(t)ĝ(-t)ṭ.

Additionally,  if f,f̂∈ L_1(;H) then 

    f(t)=f̂̂̂(-t)  (t∈).




Let f∈𝒮(;H).
By exer:derivative-under-integral we have 

    f̂ '(s)=1/√(2π)∫_(-ı t)^-ı stf(t)ṭ=-ı(̂t̂↦̂ ̂t̂f̂(̂t̂)̂)̂(s)  (s∈)

and 

    sf̂(s)=ı/√(2π)_(-ı s)^-ı stf(t)ṭ=-ıf̂'̂(s)  (s∈).

Using these formulas, one can show that f̂∈𝒮(;H).
Since the bijectivity of the Fourier transformation on 𝒮(;H)
would follow from eq:Fourier-inversion, it suffices to
prove the formulas eq:Fourier-adjoint and eq:Fourier-inversion.
Let f,g∈ L_1(;H). Then we compute using prop:interchange_integral
and Fubini's theorem 

    ∫_f̂(t)g(t)ṭ    =∫_1/√(2π)∫_^-ı stf(s)ṣg(t)ṭ
        =∫_∫_1/√(2π)^ı stf(s)g(t)ṣṭ
        =∫_f(s)1/√(2π)∫_^ı stg(t)ṭṣ
        =∫_f(s)ĝ(-s)ṣ,

which yields eq:Fourier-adjoint. For proving formula eq:Fourier-inversion,
we consider the function γ defined by γ(t)^-t^2/2 for t∈.
Clearly, γ∈𝒮(). We claim that γ̂=γ.
Indeed, we observe that γ solves the initial value problem
y'+ty=0 subject to y(0)=1; if we can show that γ̂
solves the same initial value problem, then their equality would follow from the uniqueness
of the solution. First, we observe that γ̂(0)=1/√(2π)∫_^-t^2/2ṭ=1.
Second, we compute using the formulas eq:derivative-of-fourier
and eq:fourier-of-derivative that

    γ̂'(s)=-ı(̂t̂↦̂ ̂t̂γ̂(̂t̂)̂)̂(s)=ıγ̂'̂(s)=-sγ̂(s)  (s∈).

Altogether, we have shown that γ̂ solves the same initial value problem
as γ and hence, γ̂=γ. Let now f∈ L_1(;H)
with f̂∈ L_1(;H), a>0 and x∈ H. Then we compute using eq:Fourier-adjoint

    ∫_f̂(t)γ(at)^ı stṭx    =∫_f̂(t)γ(at)x^-ı stṭ
     =∫_f(t)(̂γ̂(̂â·̂)̂x̂^̂-̂î ̂ŝ(̂·̂)̂)̂(-t)ṭ
        =∫_f(t)1/√(2π)∫_γ(ar)x^-ı sr^ı trṛṭ
        =1/a∫_f(t)γ̂(s-t/a)xṭ
     =1/a∫_f(t)γ(s-t/a)xṭ
        =∫_f(s-at)γ(t)xṭ
     =∫_f(s-at)γ(t)ṭx

for each s∈. Since this holds for all x∈ H we get 

    ∫_f̂(t)γ(at)^ı stṭ=∫_f(s-at)γ(t)ṭ  (s∈).

Letting a→0 in the latter equality, we obtain 

    ∫_f̂(t)^ı stṭ=lim_a→0∫_f(s-at)γ(t)ṭ  (s∈),

where we have used dominated convergence for the term on the left-hand
side. In order to compute the limit on the right-hand side, we first
observe that 

    ∫_∫_f(s-at)γ(t)ṭṣ≤∫_∫_f(s-at)ṣ γ(t)ṭ=f_1γ_1,

and hence, for each a>0 the operator

    S_a L_1(;H)   → L_1(;H),
    
    f    ↦(s↦∫_f(s-at)γ(t)ṭ)

is bounded by γ_1. Moreover, since S_aψ→ψ(·)γ_1
as a→0 for ψ∈(;H), we infer that 

    S_af→ f(·)γ_1  (a→0)

for each f∈ L_1(;H). Hence, passing to a suitable sequence
(a_n)_n in _>0 tending to 0, we get 

    lim_n→∞(S_a_nf)(s)→ f(s)γ_1  (a.e.  s∈).

Using this identity for the right-hand side of eq:limit_gamma,
we get 

    ∫_f̂(t)^ı stṭ=f(s)γ_1  (a.e.  s∈),

and since γ_1=√(2π), we derive eq:Fourier-inversion. 

With these preparations at hand, we are now able to prove the main
theorem of this section. 

 The mapping 

    ℱ𝒮(;H)⊆Ł(;H)→Ł(;H), f↦f̂

extends to a unitary operator on Ł(;H), again denoted by ℱ,
the Fourier transformationFourier transformation. Moreover, ℱ^∗=ℱ^-1
is given by f↦f̂(-·). 



Using eq:Fourier-adjoint and eq:Fourier-inversion
we obtain that 

    f̂ĝ_2    =∫_f̂(t)ĝ(t)ṭ
     =∫_f(t)ĝ̂(-t)ṭ
     =∫_f(t)g(t)ṭ
     =fg_2

for all f,g∈𝒮(;H) and thus, in particular, 

    f_2=ℱf_2.

Moreover, (ℱ)=(ℱ)=𝒮(;H)
is dense in Ł(;H) and hence, the first assertion follows by exer:unitary_extension. As ℱ is unitary, we have ℱ^*=ℱ^-1, thus, by eq:Fourier-adjoint applied to f,g∈𝒮(;H), we read off (using prop:adjoint_core) that ℱ^-1=(f↦f̂(-·)), which yields all the claims of the theorem at hand.


We emphasise that for f∈Ł(;H) the Fourier transform ℱf
is not given by the integral expression for L_1-functions, simply
because the integral does not need to exist. However, by dominated
convergence 

    ℱf=lim_R→∞1/√(2π)∫_-R^R^-ı t(·)f(t)ṭ,

where the limit is taken in Ł(;H). 





§ THE FOURIER–LAPLACE TRANSFORMATION AND ITS RELATION TO THE TIME
DERIVATIVE


We now use the Fourier transformation to define an analogous transformation
on our exponentially weighted Ł-type spaces; the so-called Fourier–Laplace
transformation. We recall from cor:nu=00003D0 that
for ν∈ the mapping 

    exp(-ν)(;H)→Ł(;H), f↦(t↦^-ν tf(t))

is unitary. In a similar fashion, we obtain that

    exp(-ν) L_1,ν(;H)→ L_1(;H), f↦(t↦^-ν tf(t))

defines an isometry. 

Let ν∈. We define the Fourier–Laplace transformationFourier–Laplace transformation as

    ℒ_ν(;H)
    →Ł(;H), f↦ℱexp(-ν)f.


We can also consider the Fourier–Laplace transformation as a mapping from L_1,ν(;H) to (;H); that is,

    ℒ_ν L_1,ν(;H)
    →(;H), f↦ℱexp(-ν)f.



Note that ℒ_ν=ℱexp(-ν) is unitary as an operator from (;H)
to Ł(;H) since it is the composition of two unitary operators. For φ∈(;H), we have
the expression 

    (ℒ_νφ)(t)=1/√(2π)∫_^-(ı t+ν)sφ(s)ṣ  (t∈),

which shows that ℒ_ν can be interpreted as a shifted
variant of the Fourier transformation, where the real part in the
exponent equals ν instead of zero. 


Our next goal is to show that the Fourier–Laplace transformation provides
a spectral representation of our time derivative, ν. 





















Let V→ be measurable.
We define the multiplication-by-V operatormultiplication-by-V operator as 

    V()(V())⊆Ł(;H)→Ł(;H), f↦(t↦ V(t)f(t))

with 

    (V())f∈Ł(;H)(t↦ V(t)f(t))∈Ł(;H).

In particular, if V is the identity on  we will just write  instead of 𝕀() and call it the multiplication-by-the-argument
operator, multiplication-by-the-argument operator.



Note that the multiplication-by-V operator is a vector-valued analogue of the multiplication operator seen in thm:mult1 and thm:mult2. The statements in these theorems generalise (easily) to the vector-valued situation at hand. 
Thus, as in thm:mult1, one shows that  is selfadjoint.
Moreover, when H≠{0}, in a similar fashion to the arguments carried out in thm:mult2 one shows that 

    σ()=.



In order to avoid trivial cases, we shall assume throughout that H≠{0}.


Let ν∈. Then 

    ν=ℒ_ν^∗(ı+ν)ℒ_ν.

In particular, 

    σ(ν)=ı t+νt∈.




We first prove the assertion for ν0 and show that 

    I_ν=ℒ_ν^∗(1/ı+ν)ℒ_ν.

The assertion will then follow by thm:mult1. Note that 1/ı+ν∈ L(Ł(;H))
by thm:mult1, and hence, both operators I_ν and ℒ_ν^∗(1ı+ν)ℒ_ν are bounded and
defined on the whole of (;H). Thus, it suffices to prove the
equality on a dense subset of (;H), like (;H).
We will just do the computation for the case when ν>0. So, let φ∈(;H)
and compute 

    (ℒ_νI_νφ)(t)     =1/√(2π)∫_^-(ı t+ν)s∫_-∞^sφ(r)ṛṣ
     =1/√(2π)∫_∫_r^∞^-(ı t+ν)sṣ φ(r)ṛ
        =1/√(2π)1/ı t+ν∫_^-(ı t+ν)rφ(r)ṛ
     =1/ı t+ν(ℒ_νφ)(t)

for t∈. For ν<0 the computation is analogous. In the case when
ν=0 we observe that 

    0    =exp(-ν)(ν-ν)exp(-ν)^-1
     =exp(-ν)ℒ_ν^∗(ı+ν-ν)ℒ_νexp(-ν)^-1
         =ℒ_0^∗(ı)ℒ_0.*





§ MATERIAL LAW OPERATORS


Using the multiplication operator representation of ν via
the Fourier–Laplace transformation, we can assign a functional calculus
to this operator. We will do this in the following and define operator-valued
functions of ν. The class of functions used for this calculus
are the so-called material laws. We begin by defining this function class. 

A mapping M(M)⊆→ L(H) is called a material lawmaterial law if


  * (M) is open and M is holomorphic (i.e., complex differentiable;
see also exer:holomorphic),

  * there exists some ν∈ such that _>ν⊆(M) and

    M_∞,_>νsup_z∈_>νM(z)<∞.


Moreover, we set

    Minfν∈_>ν⊆(M)  and M_∞,_>ν<∞

to be the abscissa of boundedness, ·abscissa of boundedness of M. 






  * Polynomials in z^-1: Let n∈_0, M_0,…,M_n∈ L(H).
Then 

    M(z)∑_k=0^nz^-kM_k  (z∈∖{0})

defines a material law with 

    M    =
    -∞,     if M_1=…=M_n=0,
    
    0,     otherwise.


  * Series in z^-1: Let (M_k)_k∈ in L(H) such that ∑_k=0^∞M_kr^-k<∞
for some r>0. Then 

    M(z)∑_k=0^∞z^-kM_k  (z∈∖{0})

defines a material law with M≤ r.

  * Exponentials: Let h∈,M_0∈ L(H) where M_00 and set 

    M(z) M_0^zh  (z∈).

Then M is a material law if and only if h≤0. In this case,
M=-∞.

  * Laplace transforms: Let ν∈ and k∈ L_1,ν() with
k⊆0. Then 

    M(z)√(2π)(ℒk)(z)∫_0^∞^-ztk(t)ṭ  (z∈_>ν)

defines a material law with M≤ν.

  *  Fractional powers: Let M_0∈ L(H), M_00, α∈
and set 

    M(z) M_0z^-α  (z∈∖0),

where we set 

    (r^ıθ)^-α r^-α^-ıαθ  (r>0,θ∈-ππ).

Then M is a material law if and only if α≥0 and 

    M    =
    -∞    if α=0,
    
    0     otherwise.




For material laws M we now define the corresponding material law
operators in terms of the functional calculus induced by the spectral
representation of ν. 

Let M(M)⊆→(H) be
a material law. Then, for ν>M, the operator 

    M(ı+ν)Ł(;H)→Ł(;H), f↦(t↦ M(ı t+ν)f(t))

is bounded. Moreover, we define the material law operatormaterial law operator 

    M(ν)ℒ_ν^∗M(ı+ν)ℒ_ν∈ L((;H))

and obtain 

    M(ν)≤M_∞,_>ν.




The proof is clear. 



  The set of material laws is an algebra and the mapping of assigning a material law to its corresponding material law operator is an algebra homomorphism in the following sense.
  For j∈{1,2} let M_j(M_j)⊆→(H) be material laws, λ∈.
  Then M_1+M_2 (with domain (M_1)∩(M_2)), λ M_1 and M_1· M_2 (with domain (M_1)∩(M_2)) are material laws as well.
  Moreover, M_1+M_2, M_1· M_2≤max{M_1,M_2}. Furthermore, if M_2(z) is a multiplication operator for all z∈(M_2), then
  for ν>max{M_1,M_2} we have
  (M_1M_2)(ν) = M_1(ν) M_2(ν) = M_2(ν) M_1(ν) = (M_2M_1)(ν).



We now revisit the material laws presented
in exa:material-laws and compute their corresponding operators,
M(ν).


  * Let n∈_0, M_0,…,M_n∈ L(H) and 

    M(z)∑_k=0^nz^-kM_k  (z∈∖{0}).

Then, for ν>0, one obviously has 

    M(ν)=∑_k=0^nν^-kM_k,

due to thm:spectral-repr-derivative.

  * Let (M_k)_k∈ in L(H) such that ∑_k=0^∞M_kr^-k<∞
for some r>0 and 

    M(z)∑_k=0^∞z^-kM_k  (z∈∖{0}).

Then, for ν>r, one has 

    M(ν)=∑_k=0^∞ν^-kM_k

again on account of thm:spectral-repr-derivative.

  *  Let h≤0,M_0∈ L(H) and 

    M(z) M_0^zh  (z∈).

Then, for ν∈, we have 

    M(ν)=M_0τ_h,

where 

    τ_h(;H)→(;H), f↦(t↦ f(t+h)).

Indeed, for φ∈(;H) we compute 

    (ℒ_νM_0τ_hφ)(t)     =1/√(2π)∫_^-(ı t+ν)sM_0φ(s+h)ṣ
        =M_01/√(2π)∫_^-(ı t+ν)(s-h)φ(s)ṣ
     =M(ı t+ν)(ℒ_νφ)(t)

for all t∈, where we have used prop:interchange_integral
in the second line. Hence, 

    M_0τ_hφ=ℒ_ν^∗M(ı+ν)ℒ_νφ=M(ν)φ

and since (;H) is dense in (;H) the assertion follows.

  * Let ν∈ and k∈ L_1,ν() with k⊆0
and 

    M(z)√(2π) (ℒk)(z)   (z∈_>ν).

Then, by exer:convolution, 

    M(μ)=k∗

for each μ>ν.

  * Let M_0∈ L(H), α>0 and 

    M(z) M_0z^-α  (z∈∖0).

Then for ν>0 we have 

    (M(ν)f)(t)=M_0∫_-∞^t1/Γ(α)(t-s)^α-1f(s)ṣ  (a.e. t∈)

for each f∈(;H); see exer:fractional. This
formula gives rise to the definition 

    (ν^-αf)(t)∫_-∞^t1/Γ(α)(t-s)^α-1f(s)ṣ  (t∈),

which is known as the fractional integral(Riemann–Liouville) fractional integral of order α.



Throughout the previous examples, the operator M(ν) did not
depend on the actual value of ν. Indeed, this is true for all
material laws. In order to see this, we need the following lemma. 

Let μ,ν∈ with μ<ν, and set Uz∈ z∈μν.
Moreover, let gU→ H be continuous and holomorphic on
U such that g(ı·+ν),g(ı·+μ)∈Ł(;H) and there
exists a sequence (R_n)_n∈ in _≥0 such that R_n→∞
and 

    ∫_μ^νg(±ı R_n+ρ)ρ̣→0  (n→∞).

Then 

    ℒ_μ^∗g(ı·+μ)=ℒ_ν^∗g(ı·+ν).




Let t∈. By Cauchy's integral theorem, we have that 

    ∫_γ_R_ng(z)^ztẓ=0,

where γ_R_n is the rectangular closed path with corners
±ı R_n+μ,±ı R_n+ν. 

Thus, we have that 

    ı∫_-R_n^R_ng(ı s+ν)^(ı s+ν)tṣ-ı∫_-R_n^R_ng(ı s+μ)^(ı s+μ)tṣ
        = -∫_μ^νg(-ı R_n+ρ)^(-ı R_n+ρ)tρ̣+∫_μ^νg(ı R_n+ρ)^(ı R_n+ρ)tρ̣.

Note that with the help of the formula for the inverse Fourier-transformation (see thm:Plancherel) and ℒ_ν^*=(ℱexp(-ν))^*=exp(-ν )^-1ℱ^* the left-hand side of eq:path is nothing but

    √(2π)ı((ℒ_ν^∗_-R_nR_ng(ı·+ν))(t)-(ℒ_μ^∗_-R_nR_ng(ı·+μ))(t)),

and hence, there is a subsequence of (R_n)_n (which we do not
relabel) such that the left-hand side of eq:path tends
to 

    √(2π)ı((ℒ_ν^∗g(ı·+ν))(t)-(ℒ_μ^∗g(ı·+μ))(t))

for almost every t∈ as n→∞. As such, all we need to show is
that the right-hand side of eq:path tends to 0 as
n→∞, which obviously follows by eq:integral_to_zero. 


Let M(M)⊆→ L(H) be
a material law. Then, for μ,ν>M and f∈(;H)∩μ(;H),
we have 

    M(ν)f=M(μ)f.

Moreover, M(ν) is causal for all ν>M. 



Let μ<ν. We prove the assertion for f=_ab· x
with a<b and x∈ H first. For ρ∈ we compute 

    (ℒ_ρf)(t)=1/√(2π)∫_a^bx^-(ı t+ρ)sṣ=1/√(2π)1/ı t+ρ(^-(ı t+ρ)a-^-(ı t+ρ)b)x  (t∈∖{0}).

Moreover, we define 

    g(z)1/√(2π)M(z)x1/z(^-za-^-zb)  (z∈_≥μ∖{0})

and prove that g satisfies the assumptions of lem:F-L idenpendent of nu.
First, we note that g is bounded on z∈μ≤ z≤ν∖{0}.
Indeed, we only need to prove that it is bounded near 0 provided
that μ≤0. To that end, we observe 

    1/z(^-za-^-zb)=^-za1-^-z(b-a)/z→ b-a  (z→0).

Thus, g is bounded near 0. In particular, z=0 is
a removable singularity and, hence, g can be extended holomorphically
to _≥μ. Moreover, for ρ≥μ we have that 

    ∫_g(ı t+ρ)^2ṭ=∫_-1^1g(ı t+ρ)^2ṭ+∫_|t|>1g(ı t+ρ)^2ṭ.

The first term on the right-hand side is finite since g is bounded,
while the second term can be estimated by 

    ∫_|t|>1g(ı t+ρ)^2ṭ≤M_∞,_>μ^2x^2(^-ρ a+^-ρ b)^2/2π∫_|t|>11/t^2+ρ^2ṭ<∞.

This proves that g(ı·+ρ)∈Ł(;H) for each ρ≥μ
and hence, particularly for ρ=μ and ρ=ν. Finally,
for ρ≥μ we have that 

    g(ı t+ρ)≤1/√(2π)M_∞,_>μx1/√(t^2+ρ^2)(^-ρ a+^-ρ b)→0  (t→∞),

which together with the boundedness of g yields eq:integral_to_zero
by dominated convergence. This shows that g satisfies the assumptions
of lem:F-L idenpendent of nu and thus 

    M(ν)f=ℒ_ν^∗g(ı·+ν)=ℒ_μ^∗g(ı·+μ)=M(μ)f.

By linearity, this equality extends to S_(;H) and so, 

    F S_(;H)→⋂_ν≥μ(;H), f↦ M(ν)f

is well-defined. Moreover, F is uniformly Lipschitz continuous
(with sup_ν≥μF^ν=M_∞,_>μ)
and hence, the assertions follow from lem:indepCaus. 




§ COMMENTS


The Fourier and the Fourier–Laplace transformation introduced in this
chapter are used to define an operator-valued functional calculus
for the time derivative, ν.  This functional calculus can be defined since the Fourier–Laplace transformation provides the unitary transformation yielding the spectral representation of the time derivative as multiplication operator. This fact was already noticed in <cit.>, which eventually led to evolutionary equations in <cit.>.

We emphasise that we have used the fundamental
property that both ℱ and ℒ_ν are unitary.
It is noteworthy that the Fourier transformation
is an isometric isomorphism on L_2(;X) if and only if X is a Hilbert space, see
<cit.>. In the Banach space-valued case one has to further
restrict the class of functions used to define a functional calculus.
For the topic of functional calculus we refer to the 21st ISem <cit.>
by Markus Haase and to his monograph, <cit.>.



§ EXERCISES

tocsectionExercises


 Let (Ω,Σ,μ) be
a measure space, X a Banach space and I⊆ an open interval.
Let g I×Ω→ X such that g(t,·)∈ L_1(μ;X)
for each t∈ I, and define 

    h     I→ X, t↦∫_Ωg(t,ω)μ̣(ω).



  * Assume that g(·,ω) is continuous for μ-almost every
ω∈Ω and let f∈ L_1(μ) such that 

    g(t,ω)≤ f(ω)  (t∈ I,ω∈Ω).

Prove that h is continuous.

  * Assume that g(·,ω) is differentiable for μ-almost
every ω∈Ω and let f∈ L_1(μ) such that 

    ∂_tg(t,ω)≤ f(ω)  (t∈ I,ω∈Ω).

Prove that h is differentiable with 

    h'(t)=∫_Ω∂_tg(t,ω)μ̣(ω).





Let H_0,H_1 be two Hilbert spaces
and U(U)⊆ H_0→ H_1 linear such that


  * (U) is dense in H_0 and (U) is dense in H_1.

  * ∀ x∈(U): Ux_H_1=x_H_0.

Show that U can be uniquely extended to a unitary operator between
H_0 and H_1. 



 Let Ω⊆ be open, X a complex
Banach space and fΩ→ X. Prove that the following statements
are equivalent:



  *  f is holomorphic.

  * For all x'∈ X' the mapping x'∘ fΩ→ is holomorphic.

  * f is locally bounded and x'∘ fΩ→ is holomorphic
for all x'∈ D, where D⊆ X' is a norming set[D⊆ X' is called a norming set for X, if x=sup_x'∈ D∖{0}1/x'x'(x)
for each x∈ X. Note that X' is norming for X by the Hahn–Banach
theorem.] for X.

  *  f is analytic, i.e. for each z_0∈Ω there is r>0
and (a_n)_n in X with z_0r⊆Ω
and 

    f(z)=∑_n=0^∞a_n(z-z_0)^n  (z∈z_0r).


Assume now that X=L(X_1,X_2) for two complex Banach spaces
X_1,X_2, let D_1⊆ X_1 be dense and D_2⊆ X_2'
norming for X_2. Prove that the statements 

<ref> to <ref> are equivalent

to


enumi4

  * f is locally bounded and Ω∋ z↦ x_2'(f(z)(x_1))∈
is holomorphic for all x_1∈ D_1 and x_2'∈ D_2.




 Let ν∈ and k∈ L_1,ν().
Prove that 

    ℒ_ν(k∗ f)=√(2π)(ℒ_νk)·(ℒ_νf)

for f∈(;H). 



 Let α>0 and define g_α(t)_0∞(t)t^α-1
for t∈. Show that g_α∈ L_1,ν() for each ν>0
and that 

    (ℒ_νg_α)(t)=1/√(2π)Γ(α)(ı t+ν)^-α.

Use this formula and exer:convolution to derive eq:fractional_int.

Hint: To compute the Fourier–Laplace transform of g_α, derive
that ℒ_νg_α solves a first order ordinary
differential equation and use separation of variables to solve this
equation. 



 Let μ,ν∈ with μ<ν and f∈(;H)∩μ(;H).
Moreover, set Uz∈μ< z<ν. Show that f∈⋂_μ<ρ<νρ(;H)∩ L_1,ρ(;H)
and that 

    U∋ z↦(ℒ_ zf)( z)

is holomorphic. 



 Let H_0,H_1 be Hilbert spaces
and T(;H_0)→(;H_1) linear and bounded. We call
T autonomousautonomous if Tτ_h=τ_hT for each h∈ (τ_h
denotes the translation operator defined in exa:material_law_revisited).
Prove that for autonomous T, the following statements are equivalent:



  * T is causal.

  * For all f∈(;H_0) with f⊆0∞
one has Tf⊆0∞.

Moreover, prove that for a material law M, the operator M(ν)
is autonomous for each ν>M. 



[heading=subbibliography]



CHAPTER: SOLUTION THEORY FOR EVOLUTIONARY EQUATIONS



In this chapter, we shall discuss and present the first major result
of this year's internet seminar: Picard's theorem on the solution theory for evolutionary equations which is
the main result of <cit.>. In order to stress the applicability
of this theorem, we shall deal with applications first and provide
a proof of the actual result afterwards. With an initial interest in applications in mind, we start off with the introduction of some
vector-analytic operators.



§ FIRST ORDER SOBOLEV SPACES


Throughout this section let Ω⊆^d be an open set.

We define 

    _(Ω)⊆Ł(Ω)    →Ł(Ω)^d
    ϕ   ↦(∂_jϕ)_j∈{1,…,d},


    _(Ω)^d⊆Ł(Ω)^d   →Ł(Ω)
    (ϕ_j)_j∈{1,…,d}   ↦∑_j∈{1,…,d}∂_jϕ_j,

and  if d=3,

    _(Ω)^3⊆Ł(Ω)^3   →Ł(Ω)^3
    (ϕ_j)_j∈{1,2,3}   ↦[ ∂_2ϕ_3-∂_3ϕ_2; ∂_3ϕ_1-∂_1ϕ_3; ∂_1ϕ_2-∂_2ϕ_1 ].

Furthermore, we put

    -_^*,  -_^*,  _^*

and

    _0-^*,  _0-^*,  _0^*.



The relations ,_0,,_0, and _0 are
all densely defined, closed linear operators.



The operators _,_ and _ are densely
defined by exer:Ccinfty_dense_hd. Thus, , and 
are closed linear operators by lem:ddc. Moreover, it follows from integration
by parts that _⊆, _⊆
and _⊆. Thus, , and  are also
densely defined. This, in turn, implies that _,_
and _ are closable by lem:ddc with respective closures _0,_0 and _0 by lem:ass. 

We shall describe the domains of these operators in
more detail in the next theorem.

 If f∈Ł(Ω) and g=(g_j)_j∈{1,…,d}∈Ł(Ω)^d then the following statements hold:


  * f∈() and g= f if and only if

    ∀ϕ∈(Ω),j∈{1,…,d}-∫_Ωf∂_jϕ=∫_Ωg_jϕ.


  * f∈(_0) and g=_0f if and only if there exists (f_k)_k in (Ω) such that f_k→ f
in Ł(Ω) and f_k→ g in Ł(Ω)^d as k→∞.

  * g∈() and f= g if and only if

    ∀ϕ∈(Ω)-∫_Ωg·ϕ=∫_Ωfϕ.


  * g∈(_0) and f=_0g if and only if there exists (g_k)_k in (Ω)^d such that g_k→ g
in Ł(Ω)^d and g_k→ f in Ł(Ω) as k→∞.

If d=3 and f,g∈Ł(Ω)^3 then the following statements hold:

enumi4

  * f∈() and g= f if and only if

    ∀ϕ∈(Ω)^3∫_Ωf·ϕ=∫_Ωg·ϕ.


  * f∈(_0) and g=_0f if and only if there exists (f_k)_k in (Ω)^3 such that f_k→ f
in Ł(Ω)^3 and f_k→ g in Ł(Ω)^3 as k→∞.



All the statements in thm:Sobolev_space are elementary
consequences of the integration by parts formula and the definitions
of the adjoint. We ask the reader to prove these statements in exer:Sobolev_elem.

We remark here that, classically, the following notation has been introduced: H(div,Omega), H_0(div,Omega)@H(,Ω), H_0(,Ω)H(curl,Omega), H_0(curl,Omega)@H(,Ω),H_0(,Ω)H^1(Omega), H^1_0(Omega)@H^1(Ω),H_0^1(Ω)

    H^1(Ω)    (),
    
    H_0^1(Ω)    (_0),
    
    H(,Ω)    (),
    
    H(,Ω)    ().

Following the rationale of appending zero as an index for H_0^1(Ω),
we shall also use

    H_0(,Ω)    (_0),
    
    H_0(,Ω)    (_0).

We do, however, caution the reader that other authors also use H_0(,Ω)
and H_0(,Ω) to denote the kernel of  and .
All the spaces just defined are so-called Sobolev spacesSobolev space. We note
that for d=3 we clearly have H^1(Ω)^3⊆ H(,Ω)∩ H(,Ω).
On the other hand, note that H(,Ω) is neither a sub-
nor a superset of H(,Ω).



  We emphasise that H_0^1(Ω)⊆ H^1(Ω) is a proper inclusion for many open Ω. The `0' in the index is a reminder of `0'-boundary conditions. In fact, the only difference between these two spaces lies in the behaviour of their elements at the boundary of Ω. The space H_0^1 signifies all H^1-functions vanishing at ∂Ω in a generalised sense. The corresponding statements are true for the inclusions H_0(,Ω)⊆ H(,Ω) and H_0(,Ω)⊆ H(,Ω). The space H_0(,Ω) describes H(,Ω)-vector fields with vanishing normal component and to lie in H_0(,Ω) provides a handy generalisation of vanishing tangential component. We will anticipate these abstractions, when we apply the solution theory of evolutionary equations for particular cases. In a later chapter we will come back to this issue when we discuss inhomogeneous boundary value problems.


For later use, we record the following relationships between the vector-analytical
operators introduced above.


 Let d=3. We have the following inclusions:

    (_0)    ⊆(_0),
    (_0)    ⊆(_0),
    ()    ⊆(),
    ()    ⊆().




It is elementary to show that for given ψ∈(Ω)^3 and
ϕ∈(Ω) we have _0_0ψ=0 as well
as _0_0ϕ=0. Thus, we obtain (_)⊆(_0) and (_)⊆(_0).
Since (_0) and (_0) are closed, and (Ω)^3
and (Ω)  are cores for _0 and _0 respectively, we
obtain the first two inclusions. The last two inclusions follow from
the first two 
by taking into account the orthogonal decompositions

    Ł(Ω)^3 =()⊕(_0)=()⊕(_0)

and

    Ł(Ω)^3 =(_0)⊕()=(_0)⊕()

which follow from cor:abstrHelmH.




§ WELL-POSEDNESS OF EVOLUTIONARY EQUATIONS AND APPLICATIONS


The solution theory of evolutionary equationsSolution Theory, evolutionary equations is contained in the
next result, Picard's theorem. This result is central for all the derivations to come.
In fact, with the notation of thm:Solution_theory_EE,
we shall prove that for all (well-behaved) F there is a unique solution
of

    (νM(ν)+A)U=F.

The solution U depends continuously and causally on the choice
of F. 



In order to formulate the result, for a Hilbert space H, ν∈ and a given operator A(A)⊆ H→ H we define its extended operator in (;H), again denoted by A, by 

    (;(A))⊆(;H)    →(;H)
    
    f    ↦(t↦ Af(t)).

We have collected some properties of extended operators in
exer:tensorprod and exer:abstractFourier_exten.


 Let ν_0∈ and H be a Hilbert
space. Let M(M)⊆→(H) be a material law
with M<ν_0 and let A(A)⊆ H→ H
be skew-selfadjoint. Assume that

    ϕzM(z)ϕ_H≥ cϕ_H^2  (ϕ∈ H,z∈_≥ν_0)

for some c>0. Then for all ν≥ν_0 the operator νM(ν)+A is closable and

    S_ν(νM(ν)+A)^-1∈((;H)).

Furthermore, S_ν is causal and satisfies S_ν_()≤1/c, and for all F∈(ν) we
have

    S_νF∈(ν)∩(A).

Furthermore, for η,ν≥ν_0 and F∈(;H)∩η(;H) we have that S_νF=S_ηF.


The property that S_νF=S_ηF for all F∈(;H)∩η(;H)
where η,ν≥ν_0, for some ν_0∈, will be referred
to as S_ν being eventually independent of ν in what follows.


 Recall that F∈(ν) implies U S_ν F∈(ν)∩(A) in thm:Solution_theory_EE. Since M(ν) leaves the space (ν) invariant, this gives that M(ν)U∈(ν) and thus, U solves the evolutionary equation literally; that is,
 
    (νM(ν)+A)U=F,

while for F∈(;H), in general, we just have

    (νM(ν)+A)U=F.




 Let H be a Hilbert space and T∈ L(H). If T is selfadjoint, we write T≥ c for some c∈ if 

    ∀ x∈ H: xTx_H≥ cx_H^2.

Moreover, we define the real part of T Real part of operator by T1/2(T+T^∗).


Note that if H is a Hilbert space and T∈ L(H) then T is selfadjoint. Moreover,

    x( T)x_H = xTx_H   (x∈ H).

Hence, in thm:Solution_theory_EE the assumption on the material law can be rephrased as 

    zM(z)≥ c   (z∈_≥ν_0).

The following operators will be prototypical examples needed for the applications
of the previous theorem.


Let H_0,H_1 be Hilbert spaces.


  *  Let B(B)⊆ H_0→ H_1, C(C)⊆ H_1→ H_0 be
densely defined linear operators. Then

    [ 0 C; B 0 ](B)×(C)⊆ H_0× H_1   → H_0× H_1
    
    (ϕ,ψ)    ↦(Cψ,Bϕ)

is densely defined, and we have

    [ 0 C; B 0 ]^*=[   0 B^*; C^*   0 ].


  *  Let a∈(H_0), and  c>0. Assume a≥ c.
Then a^-1∈(H_0) with a^-1≤1/c and a^-1≥ ca^-2.




The proof of the first statement can be done in two steps. First, notice that the inclusion
[   0 B^*; C^*   0 ]⊆[ 0 C; B 0 ]^* follows immediately. If, on the other hand, [ ϕ; ψ ]∈([ 0 C; B 0 ]^*) with [ 0 C; B 0 ]^*[ ϕ; ψ ]=[ ξ; ζ ] we get for all x∈(B) that

    Bxψ_H_1    =[ 0 C; B 0 ][ x; 0 ][ ϕ; ψ ]_H_0× H_1
    =[ x; 0 ][ 0 C; B 0 ]^*[ ϕ; ψ ]_H_0× H_1
        =[ x; 0 ][ ξ; ζ ]_H_0× H_1=xξ_H_0.

Hence, ψ∈(B^*) and B^*ψ=ξ. Similarly, we obtain
ϕ∈(C^*) and C^*ϕ=ζ.

For the second statement, we compute for all ϕ∈ H_0 using
the Cauchy–Schwarz inequality

    ϕaϕ_H_0≥ϕaϕ_H_0≥ϕaϕ_H_0≥ cϕϕ_H_0 = cϕ_H_0^2.

Thus, a is one-to-one. Since a= a^* it follows that
a^* is one-to-one, as well. Thus, we get that a has dense
range by thm:ran-kernerl. The inequality 

    aϕ_H_0≥ cϕ_H_0

implies that a^-1 is bounded with a^-1≤1/c. Hence, as a^-1 is closed, (a^-1)=(a) is closed by lem:bbc and hence, (a^-1)=H_0; that is, a^-1∈(H_0).
To conclude, let ψ∈ H_0 and put ϕ a^-1ψ.
Then ψ_H_0=aa^-1ψ_H_0≤aa^-1ψ_H_0
and so

    ψa^-1ψ_H_0    = aϕϕ_H_0
    = ϕaϕ_H_0≥ cϕϕ_H_0
     = ca^-1ψa^-1ψ_H_0
       ≥ c1/a^2ψ_H_0^2.*





 §.§ The Heat Equation


heat equation, evolutionary equation
The first example we will consider is the heat equation in an open subset Ω⊆^d. Under a heat source,
Q×Ω→, the heat distribution, θ×Ω→,
satisfies the so-called heat-flux-balanceheat-flux-balance

    ∂_tθ+ q=Q.

Here, q×Ω→^d is the heat fluxheat flux which is connected
to θ via Fourier's lawFourier's law

    q=-aθ,

where aΩ→^d× d is the heat conductivity,
which is measurable, bounded and uniformly strictly positive in the
sense that 

    a(x)≥ c

for all x∈Ω and some c>0 in the sense of positive definiteness. Moreover,
we assume that Ω is thermally isolated, which is modelled
by requiring that the normal component of q vanishes at ∂Ω; that
is, q∈(_0). Written as a block matrix and incorporating
the boundary condition, we obtain 

    (∂_t[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0 _0;     0 ])[ θ; q ]=[ Q; 0 ].



For all ν>0, the operator 

    ν[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0 _0;     0 ]

is densely defined and closable in (;Ł(Ω)×Ł(Ω)^d).
The respective closure is continuously invertible with causal inverse
being eventually independent of ν. 



The assertion follows from thm:Solution_theory_EE applied
to 

    M(z)=[ 1 0; 0 0 ]+z^-1[    0    0;    0 a^-1 ]  and   
    A=[  0 _0;     0 ].

Note that M is a material law with M=0 by exa:material-laws. Moreover, for (x,y)∈Ł(Ω)×Ł(Ω)^d and z∈_≥ν with ν>0 we estimate 

    (x,y)zM(z)(x,y)_Ł(Ω)×Ł(Ω)^d   ≥ zx_Ł(Ω)^2+ ca^-2y_Ł(Ω)^d^2
       ≥min{ν ,ca^-2}(x,y)^2_Ł(Ω)×Ł(Ω)^d,

where we have used  prop:block_op_realinv<ref> in the first inequality.
Moreover, A is skew-selfadjoint by prop:block_op_realinv<ref>.



Assume that Q∈(ν). It then follows from thm:Solution_theory_EE
that 

    [ θ; q ]   (ν[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0 _0;     0 ])^-1[ Q; 0 ]
       ∈(ν)∩([  0 _0;     0 ]).

Then, as in rem:regularity, it follows that θ and q satisfy the heat-flux-balance
and Fourier's law in the sense that θ∈(ν)∩()
and q∈(_0) and

    ∂_tθ+_0q     =Q,
    
    q     =-aθ.

This regularity result is true even for Q∈(;Ł(Ω));
see <cit.>.





 §.§ The Scalar Wave Equation


wave equation, scalar, evolutionary equation
The classical scalar wave equation in a medium Ω⊆^d
(think, for instance, of a vibrating string (d=1) or membrane (d=2))
consists of the equation of the balance of momentumbalance of momentum where the acceleration
of the (vertical) displacement, u×Ω→, is balanced
by external forces, f×Ω→, and the divergence
of the stressstress, σ×Ω→^d, in such a  way that

    ∂_t^2u-σ=f.

The stress is related to u via the following so-called stress-strain
relation (here Hooke's law)Hooke's law

    σ=T u,

where the so-called elasticity tensorelasticity tensor, TΩ→^d× d, is bounded, measurable, and satisfies

    T(x)=T(x)^*≥ c

for some c>0 uniformly in x∈Ω. The quantity u
is referred to as the strain. We think of u as being fixed at ∂Ω
(“clamped boundary condition”).clamped boundary condition This is modelled by u∈(_0). 

Using v∂_tu as an unknown, we can rewrite the
balance of momentum and Hooke's law as 2×2-block-operator
matrix equation

    (∂_t[    1    0;    0 T^-1 ]-[  0   ; _0  0 ])[ v; σ ]=[ f; 0 ].

The solution theory of evolutionary equations for the wave equation
now reads as follows:

 Let Ω⊆^d be open, and T as
indicated above. Then, for all ν>0, 

    ν[    1    0;    0 T^-1 ]-[  0   ; _0  0 ]

is densely defined and closable in (;Ł(Ω)×Ł(Ω)^d).
The respective closure is continuously invertible with causal inverse
being eventually independent of ν. 



We apply thm:Solution_theory_EE to A=-[  0   ; _0  0 ], which is skew-selfadjoint by prop:block_op_realinv<ref>,
and M(z)=[    1    0;    0 T^-1 ], which defines a material law with M=-∞. The positive
definiteness constraint needed in thm:Solution_theory_EE
is satisfied by prop:block_op_realinv<ref> on account of the selfadjointness
of T, which implies the same for T^-1. Indeed, for ν_0 >0 and z∈_≥ν_0 we estimate

    (x,y)zM(z)(x,y)_Ł(Ω)×Ł(Ω)^d   = xz x_Ł(Ω)+yzT^-1y_Ł(Ω)^d
       ≥ν_0 x^2_Ł(Ω) + ν_0 c/T^2y_Ł(Ω)^d^2
       ≥ν_0 min{1,c/T^2}(x,y)_Ł(Ω)×Ł(Ω)^d^2

for each (x,y)∈Ł(Ω)×Ł(Ω)^d, where we used the selfadjointness of T^-1 in the second line.



Let f∈(;Ł(Ω)), ν>0, and define

    [  u; σ̃ ]=(ν[    1    0;    0 T^-1 ]-[  0   ; _0  0 ])^-1[ ν^-1f;     0 ].

By thm:Solution_theory_EE, we obtain [  u; σ̃ ]∈(ν)∩([  0   ; _0  0 ]). Hence, we have

    νu-σ̃    =ν^-1f
    νT^-1σ̃    =_0u

or

    νu-σ̃    =ν^-1f
    σ̃    =Tν^-1_0u.

Thus, formally, after another time-differentiation and the setting of
σ=νσ̃ we obtain a solution of the wave equation,
(u,σ). Notice, however, that differentiating σ̃
cannot be done without any additional knowledge of the regularity
of σ̃. In fact, in order to arrive at the balance of
momentum equation, one would need to have σ̃∈(ν).
However, one only has σ̃∈(ν)∩().
It is an elementary argument, see <cit.>, that
we in fact have ν^-1=ν^-1,
which suggests that, in general, σ̃∉(ν), see exer:commute_integral.





 §.§ Maxwell's Equations


Maxwell's equations, evolutionary equation
The final example in this lecture forms the archetypical evolutionary
equation – Maxwell's equations in a medium Ω⊆^3.
In order to see this (and to finally conclude the 2×2-block
matrix formulation historically due to the work of <cit.>), we start out with Faraday's lawFaraday's law of induction,
which relates the unknown electric field, E×Ω→^3,
to the magnetic induction, B×Ω→^3, via

    ∂_tB+ E=0.

We assume that the medium is contained in a perfect conductor, which is reflected
in the so-called electric boundary conditionelectric boundary condition which asks for the vanishing of the
tangential component of E at the boundary. This is modelled by E∈(_0).
The next constituent of Maxwell's equations is Ampere's lawAmpere's law

    ∂_tD+J_c- H=J_0,

which relates the unknown electric displacementelectric displacement, D×Ω→^3,
chargecharge, J_c×Ω→^3, and magnetic fieldmagnetic field, H×Ω→^3,
to the (given) external currents, external current J_0×Ω→^3.
Maxwell's equations are completed by constitutive relations specific
to each material at hand. Indeed, the (bounded, measurable) dielectricitydielectricity ε,
εΩ→^3×3, and the (bounded, measurable)
magnetic permeabilitymagnetic permeability μ, μΩ→^3×3, are symmetric matrix-valued
functions which couple the electric displacement electric displacement to the electric field electric field
and the magnetic field magnetic field to the magnetic induction magnetic induction via

    D=ε E, and  B=μ H.

Finally, Ohm's law Ohm's law relates the charge to the electric field via the
(bounded, measurable) electric conductivityelectric conductivity σ, σΩ→^3×3,
as

    J_c=σ E.

All in all, in terms of (E,H),  Maxwell's equations read 

    (∂_t[ ε 0; 0 μ ]+[ σ 0; 0 0 ]+[  0  -; _0  0 ])[ E; H ]=[ J_0;   0 ].

For the time being, we shall assume that there exists c>0 and ν_0>0
such that for all ν≥ν_0 we have

    νε(x)+σ(x)≥ c,  μ(x)≥ c   (x∈Ω)

in the sense of positive definiteness. Note that the latter condition
allows particularly for ε=0 on certain regions, if σ
compensates for this. This situation is referred to as the eddy current
approximation in these regions. With the above preparations at hand, we may now formulate
the well-posedness result concerning Maxwell's equations.

 Let Ω⊆^3 be open and ν≥ν_0.
Then

    ν[ ε 0; 0 μ ]+[ σ 0; 0 0 ]+[  0  -; _0  0 ]

is densely defined and closable in (;Ł(Ω)^3×Ł(Ω)^3).
The respective closure is continuously invertible with causal inverse
being eventually independent of ν. 



The assertion follows from thm:Solution_theory_EE applied
to the material law 

    M(z)=[ ε 0; 0 μ ]+z^-1[ σ 0; 0 0 ]

and the skew-selfadjoint operator 

    A=[  0  -; _0  0 ].



In the physics literature (see e.g. <cit.>), Maxwell's equations are usually
complemented by Gauss' law,

    _0B=0,

as well as the introduction of the charge density, ρ=ε E, and
the current, J=J_0-J_c, by the continuity equation

    ∂_tρ= J.

We shall argue in the following that these equations are automatically
satisfied if (E,H) is a solution to Maxwell's equation. Indeed,
assuming J_0∈(ν), then, as a consequence of thm:Solution_theory_EE,
we have that 

    [ E; H ]    = (ν[ ε 0; 0 μ ]+[ σ 0; 0 0 ]+[  0  -; _0  0 ])^-1[ J_0;   0 ]
         ∈(ν)∩([  0  -; _0  0 ]).

Reformulating the latter equation yields

    B     =μ H=-ν^-1_0E,
    ε E     =ν^-1(-σ E+J_0+ H)=ν^-1J+ν^-1 H.

Since _0E∈(_0), we have by prop:interchange_integral
that ν^-1_0E∈(_0). Thus, by prop:ker_ran_vec_ana,
we obtain

    _0B=_0(-ν^-1_0E)=0.

Similarly, we deduce that 

    ρ=ε E=ν^-1J.

If, in addition, we have that J∈(), we recover the continuity
equation. In general, the continuity equation is satisfied in the
integrated sense just derived.


We shall keep the list of examples to that for now. In the course
of this internet seminar, we will see more (involved) examples. Furthermore,
we will study the boundary conditions more deeply and shall relate
the conditions introduced abstractly here to more classical formulations
involving trace spaces.



§ PROOF OF PICARD'S THEOREM


In this section we shall prove the well-posedness theorem. For this,
we recall an elementary result from functional analysis. It is remindful
of the Lax–Milgram lemma.Lax–Milgram lemma

Let H be a Hilbert space and B(B)⊆ H→ H
densely defined and closed with (B)⊇(B^*). Assume
there exists c>0 such that 

    ϕBϕ_H≥ cϕ_H^2  (ϕ∈(B)).

Then B^-1∈(H) and B^-1≤1/c.



The proof is a refinement of the argument in prop:block_op_realinv.
In fact, the assumed inequality implies closedness of the range of B as
well as continuous invertibility with B^-1(B)→ H.
The fact that (B) is dense in H follows from the fact that
ϕB^*ϕ_H≥ cϕ_H^2 for all ϕ∈(B^*)⊆(B)
which, in turn, also follows from the assumed inequality.



Let ν≥ν_0 and 











  z∈_≥ν. Define B(z) zM(z)+A. Since
M(z)∈(H) it follows from thm:adj-sum that B(z)^*=(zM(z))^*-A
and (B(z))=(B(z)^*)=(A). Moreover, for all ϕ∈(A)
we have

    ϕB(z)ϕ_H=ϕ(zM(z)+A)ϕ_H=ϕzM(z)ϕ_H≥ cϕ_H^2,

due to the skew-selfadjointness of A. Thus, by prop:accr_inv applied to B(z) instead of
B, we deduce that 

    S_≥ν∋ z↦ B(z)^-1

is bounded  and assumes values in (H) with norm bounded by 1/c.
By exer:comp_ana, we have that S is holomorphic. Thus,
S is a material law and S(ν)≤1/c by prop:mat_law_function_of_td.
Moreover, thm:material law independent of nu implies
that S(ν) is independent of ν and causal.

Next, if f∈(ν), it follows that (ı+ν)ℒ_νf∈Ł(;H).
Hence, for all t∈ we obtain 

    AS(ı t+ν)ℒ_νf(t)     =A((ı t+ν)M(ı t+ν)+A)^-1ℒ_νf(t)
        =ℒ_νf(t)-(ı t+ν)M(ı t+ν)S(ı t+ν)ℒ_νf(t).

Thus, by the boundedness of M and S, we deduce S(ı·+ν)ℒ_νf∈Ł(;(A)).
This implies S(ν)f∈(;(A)) by exer:abstractFourier_exten.
Similarly, but more easily, it follows that (ı·+ν)S(ı·+ν)ℒ_νf∈Ł(;H) also, which shows S(ν)f∈(ν).

We now define the operator B(ı+ν) by

    (B(ı+ν)){f∈Ł(;H) ;     f(t)∈(A)  t∈,
        (t↦ B(ı t+ν)f(t))∈Ł(;H)}

and

    B(ı+ν)f (t↦ B(ı t+ν)f(t))   (f∈(B(ı+ν))).

Then one easily sees that B(ı+ν)=S(ı+ν)^-1 and since S(ı+ν) is closed, it follows that B(ı+ν) is closed as well. Moreover

    (ı+ν)M(ı+ν)+A⊆ B(ı+ν)

and hence, the operator (ı+ν)M(ı+ν)+A is closable, which also yields the closability of νM(ν)+A by unitary equivalence. To complete the proof, we have to show that

    (ı+ν)M(ı+ν)+A=B(ı+ν),

as this equality would imply S(ν)=(νM(ν)+A)^-1 by unitary equivalence. For showing the asserted equality, let f∈(B(ı+ν)). For n∈ we define f_n_-nn f. Then f_n∈(ı+ν)∩(A) ⊆((ı+ν)M(ı+ν)+A) for each n∈ and by dominated convergence, we have that f_n→ f as n→∞ as well as

    ((ı+ν)M(ı+ν)+A)f_n=B(ı+ν)f_n=_-nn B(ı+ν)f→ B(ı+ν)f   (n→∞).

This shows that f∈((ı+ν)M(ı+ν)+A) and hence, the assertion follows.



  Note that thm:Solution_theory_EE can partly be generalised in the following way (with the same proof). Let M_>ν_0→(H) be holomorphic and A a closed, densely defined operator in H
  such that zM(z)+A is boundedly invertible for all z∈_>ν_0 and that sup_z∈_>ν_0(zM(z)+A)^-1_(H)<∞. Then S_ν∈((;H)) is causal and eventually independent of ν.



  As the proof of thm:Solution_theory_EE shows, for ν≥ν_0 we have that S_≥ν∋ z↦ (zM(z)+A)^-1∈(H) is a material law and S_ν = S(ν). Thus, the solution operator is a material law operator, and by rem:mlohom applied to S and z↦1/z1_H we obtain 
  
    S_νν⊆ν S_ν.








§ COMMENTS




The proof of thm:Solution_theory_EE here is rather close to the strategy originally employed in <cit.>, at least where existence and uniqueness are concerned. The causality part is a consequence of some observations detailed in <cit.>. The original process of proving causality used the Theorem of Paley and Wiener, which we shall discuss later on.

The eddy current approximation eddy current approximation has enjoyed great interest in the mathematical and physical community,
in particular for the case when ε=0 everywhere. The reason
being that then Maxwell's equations are merely of parabolic type. We
shall refer to <cit.> and the references therein for an
extensive discussion. 

Both prop:accr_inv and the Lax–MilgramLax–Milgram lemma
lemma have been put into a general perspective in <cit.>.




§ EXERCISES

tocsectionExercises


Let (Ω,Σ,μ) be a
σ-finite measure space and let H_0,H_1 be Hilbert spaces. Let A(A)⊆ H_0→ H_1
be densely defined and closed. Show that the operator

    A_μŁ(μ;(A))⊆Ł(μ;H_0)    →Ł(μ;H_1)
    
    f    ↦(ω↦ Af(ω))

is densely defined and closed. Moreover, show that (A_μ)^*=(A^*)_μ.



In the situation of exer:tensorprod,
if (Ω_1,Σ_1,μ_1) is another σ-finite measure space and ℱŁ(μ)→Ł(μ_1)
is unitary, show that for j∈{0,1} there exists a unique unitary operator ℱ_H_jŁ(μ;H_j)→Ł(μ_1;H_j)
such that 

    ℱ_H_j (ϕ x)=(ℱϕ) x   (ϕ∈Ł(μ),x∈ H_j).

Furthermore, prove that

    ℱ_H_1A_μℱ_H_0^*=A_μ_1.




Show that for Ω⊆^d
open, the set (Ω)⊆Ł(Ω) is dense. 



Prove thm:Sobolev_space.



Let H be a Hilbert space, A(A)⊆ H→ H
skew-selfadjoint, and c>0. Moreover, let M(M)⊆→(H) be holomorphic with 

    M(z)≥ c    (z∈ (M)).

Show that (M)∋ z↦(M(z)+A)^-1 is holomorphic.



 Let C(C)⊆ H_0 → H_1 be a densely defined and closed linear operator acting in Hilbert spaces H_0 and H_1. For ν>0 show that

    ν^-1C=Cν^-1.

Hint: Apply exer:abstractFourier_exten and show (ı+ν)^-1C=C(ı+ν)^-1 with a suitable approximation argument.






  * Compute H_0^1(Ω)^ where the orthogonal complement
is computed in H^1(Ω).

  * Assume that

    Dϕ∈ H^1(Ω)ϕ∈(),ϕ=ϕ⊆ C^∞(Ω).
 and show that C^∞(Ω)∩ H^1(Ω)⊆ H^1(Ω)
is dense.

Remark: The regularity assumption in (b) always holds and is known as Weyl's Lemma, see e.g. <cit.>, where the more general situation of an elliptic operator with smooth coefficients is treated. See also <cit.>, where the regularity is shown for harmonic distributions.



[heading=subbibliography]




CHAPTER: EXAMPLES OF EVOLUTIONARY EQUATIONS


This chapter is devoted to a small tour through a variety of evolutionary
equations. More precisely, we shall look into the equations of poro-elastic
media, (time-)fractional elasticity, thermodynamic media with delay
as well as visco-elastic media. The discussion of these examples will
be similar to that of the examples in the previous chapter
in the sense that we shall present the equations first, reformulate
them suitably and then apply the solution theory to them. The
study of visco-elastic media within the framework of partial integro-differential
equations will be carried out in the exercises section.



§ PORO-ELASTIC DEFORMATIONS


In this section we will discuss the equations of poro-elasticityporo-elasticity, which form a coupled system of equations. More precisely, the equations of (linearised)
elasticity are coupled with the diffusion equation. Before properly
writing these equations we introduce the following notation and differential operators.

  Let ^d× d_A ∈^d× dA=A^⊤⊆^d× d the (closed) subspace of symmetric d× d matrices. 
  Let Ω⊆^d be open. Then define
  
    Ł(Ω)_^d× d   Ł(Ω; ^d× d_) 
        = (Φ_jk)_j,k∈{1,…,d}∈Ł(Ω)^d× d∀ j,k∈{1,…,d}Φ_jk=Φ_kj.

  Analogously, we set
  (Ω)_^d× d(Ω;^d× d_).

Note that the symmetry of a d× d matrix here means that the matrix elements are symmetric with respect to the main diagonal. For =, this does not correspond to the symmetry of the associated linear operator (which would rather be selfadjointness).


Let Ω⊆^d be open. Then we define

    _(Ω)^d⊆Ł(Ω)^d   →Ł(Ω)_^d× d
    (ϕ_j)_j∈{1,…,d}   ↦1/2(∂_kϕ_j+∂_jϕ_k)_j,k∈{1,…,d},

and

    _(Ω)_^d× d⊆Ł(Ω)_^d× d   →Ł(Ω)^d
    (Φ_jk)_j,k∈{1,…,d}   ↦(∑_k=1^d∂_kΦ_jk)_j∈{1,…,d}.

Similarly to the definitions in the previous chapter, we put -_^*,
-_^* and _0-^*,
_0-^*, where (analogously to the scalar-valued case)
we observe that _⊆-_^* motivating the
notation  and _0. 



Note that in the literature u is also denoted by ε(u)
and is called the strain tensorstrain tensor. Due
to the (obvious) similarity to the scalar case, we refrain from using
ε in this context and prefer  instead. Again,
the index 0 in the operators refers to generalised Dirichlet (for _0) or
Neumann (for _0) boundary conditions.


We are now properly equipped to formulate the equations of poro-elasticityporo-elasticity;
see also <cit.> and below for further details. In an elastic
body Ω⊆^d, the displacement field, u×Ω→^d,
and the pressure field, p×Ω→, of a fluid
diffusing through Ω satisfy the following two energy balance
equations 

    ∂_tρ∂_tu-∂_tλ u- C u+α^*p     =f,
    ∂_t(c_0p+α u)- k p     =g.

The right-hand sides f×Ω→^d and g×Ω→
describe some given external forcing. We assume homogeneous Neumann
boundary conditions for the diffusing fluid as well as homogeneous
Dirichlet (i.e. clamped) boundary conditions for the elastic body.
The operator ρ∈(Ł(Ω)^d) describes the density of
the medium Ω (usually realised as a multiplication operator by a bounded, measurable, scalar
function). The bounded linear operators C∈(Ł(Ω)_^d× d)
and k∈(Ł(Ω)^d) are the elasticity tensor and the hydraulic
conductivity of the medium, whereas c_0,λ∈(Ł(Ω))
are the porosity of the medium and the compressibility of the fluid,
respectively. The operator α∈(Ł(Ω)) is the so-called
Biot–Willis constant. Note that in many applications ρ,c_0,λ and α
are just positive real numbers, and C and k are strictly positive
definite tensors or matrices.

The reformulation of the equations for poro-elasticity involves several
`tricks'. One of these is to introduce the matrix trace as the operator

    Ł(Ω)_^d× d   →Ł(Ω)
    
    (Φ_jk)_j,k∈{1,…,d}   ↦∑_j=1^dΦ_jj.

Note that the adjoint is given by ^*f=(f,f,…,f)∈Ł(Ω)_^d× d. 
It is then elementary to obtain ⊆ as well
as =^*. Hence, we get formally

    ∂_tρ∂_tu-((∂_t^*λ+C) u-^*α^*p)     =f,
    ∂_t(c_0p+α u)- k p     =g.

Next, we introduce a new set of unknowns

    v    ∂_tu,
    
    T     C u,
    ω   λ v-α^*p,
    
    q    -k p.

Here, v is the velocity, T is the stress tensor and q is the heat flux. The quantity ω is an additional variable, which helps to rewrite the system into the form of evolutionary equations.

In order to finalise the reformulation we shall assume some additional properties
on the coefficients involved. Throughout the rest of this section,
we assume that

    ρ=ρ^*   ≥ c,
    
    c_0=c_0^*   ≥ c,
    λ   ≥ c,
     k    ≥ c, and 
    
    C=C^*   ≥ c

for some c>0, where all inequalities are thought of in the sense of positive definiteness (compare Chapter <ref>). As a consequence, we obtain

    v=λ^-1ω+λ^-1α^*p.

Rewriting the defining equations for T, ω, and q
together with the two equations we started out with, we obtain the
system

    ∂_tρ v-(T+^*ω)     =f,
    ∂_tc_0p+αλ^-1ω+αλ^-1α^*p+ q     =g,
    λ^-1ω+λ^-1α^*p- v     =0,
    ∂_tC^-1T- v     =0,
    
    k^-1q+ p     =0.

Note that at this stage of modelling we did assume that we can freely interchange the order of differenation, so that
∂_t u = ∂_t  u.
Introducing 

    M_0   [    ρ    0    0    0    0;    0  c_0    0    0    0;    0    0    0    0    0;    0    0    0 C^-1    0;    0    0    0    0    0 ],   
    M_1[        0        0        0        0        0;        0 αλ^-1α^*    αλ^-1        0        0;        0  λ^-1α^*     λ^-1        0        0;        0        0        0        0        0;        0        0        0        0     k^-1 ],
    
    V    [ 1 0 0 0 0; 0 1 0 0 0; 0 0 1   0; 0 0 0 1 0; 0 0 0 0 1 ],   
    A [   0   0   0   -   0;   0   0   0   0  _0;   0   0   0   0   0; -_0   0   0   0   0;   0       0   0   0 ],

we obtain 

    (∂_tM_0+M_1+VAV^*)[ v; p; ω; T; q ]=[ f; g; 0; 0; 0 ].

This perspective enables us to prove well-posedness for the equations
of poro-elasticity by applying thm:Solution_theory_EE.


 Put HŁ(Ω)^d×Ł(Ω)×Ł(Ω)×Ł(Ω)_^d× d×Ł(Ω)^d
and let M_0,M_1,V∈(H) and A be given as in eq:PoroM and eq:PoroVA.
Then there exists ν_0>0 such that for all ν≥ν_0
the operator νM_0+M_1+VAV^*
is continuously invertible on (;H). The inverse
S_ν of this operator is causal and eventually independent of ν. Moreover,
sup_ν≥ν_0S_ν<∞ and F∈(ν)
implies S_νF∈(ν)∩(VAV^*).


We will provide two prerequisites for the proof. We ask for the details of the proof in exer:Poro. 


 Let H_0, H_1 be Hilbert spaces, B(B)⊆ H_0→ H_0
skew-selfadjoint, V∈(H_0,H_1) bijective.
Then (VBV^*)^*=-VBV^*.


The proof of prop:UAU is left as (part of) exer:Poro.


 Let H be a Hilbert space, N_0,N_1∈(H)
with N_0=N_0^*. Assume there exist c_0,c_1>0 such that xN_0x≥ c_0x^2 for all x∈(N_0)
and yN_1y≥ c_1y^2 for all y∈(N_0). 
Then for all 0<c_1'<c_1 there exists ν_0>0 such that
for all ν≥ν_0 we have that

    ν N_0+ N_1≥ c'_1.




Note that by the selfadjointness of N_0 we can decompose H=(N_0)⊕(N_0), see cor:abstrHelmH. 
Let z∈ H, and x∈(N_0), y∈(N_0) such that z=x+y. For ε,ν>0 we estimate

    νx+yN_0(x+y)+x+yN_1(x+y)
        =νxN_0x+yN_1y+xN_1x+xN_1y+yN_1x
       ≥ν c_0x^2+c_1y^2-N_1x^2-2N_1xy
       ≥ν c_0x^2+c_1y^2-N_1x^2-1/εN_1^2x^2-εy^2
        =(ν c_0-1/εN_1^2-N_1)x^2+(c_1-ε)y^2,

where we have used the Peter–Paul inequality (i.e., Young's inequality for products of non-negative numbers).
For 0<c_1'<c_1 we find ε>0 such that c_1-ε>c_1'.
Then we choose ν>1/c_0(c_1'+1/εN_1^2+N_1).
With this choice of ν_0 we deduce for all ν≥ν_0 that

    νzN_0z+zN_1z≥ c_1'(x^2+y^2)=c_1'z^2,

which yields the assertion.




§ FRACTIONAL ELASTICITYFRACTIONAL ELASTICITY


Let Ω⊆^d be open.
In order to better fit to the experimental data of visco-elastic solids (i.e., to incorporate solids that `memorise' previous force applied to them) the equations of linearised elasticity need to be extended in some way. The balance law for
the momentum, however, is still satisfied; that is, for the displacement
u×Ω→^d we still have that

    ∂_tρ∂_tu- T=f,

where ρ∈(Ł(Ω)^d) models the density and f×Ω→^d
is a given external forcing term. The stress tensor, T×Ω→_sym^d× d,
does not follow the classical Hooke's law, which, if it did, would look like 

    T=C u

for C∈(Ł(Ω)_sym^d× d). Instead it is
amended by another material dependent coefficient D∈(Ł(Ω)_sym^d× d)
and a fractional time derivative; that is,

    T=C u+D∂_t^α u,

for some α∈01, where ∂_t^α∂_t ∂_t^α-1, see exa:material-laws<ref>. We shall simplify the present consideration
slightly and refer to exer:fracEla instead for a more
involved example. Throughout this section, we shall assume that 

    C=0,  D=D^*≥ c, and ρ=ρ^*≥ c

for some c>0. Thus, putting v∂_tu and assuming
the clamped boundary conditions again, we study well-posedness of

    ∂_tρ v- T     =f,
    
    T     =D∂_t^α_0u.



In order to do that, we first rewrite the second equation. 
We will make use of the following proposition which will serve us to show bounded invertibility of ∂_t^α (in the space ), and which will also be employed to obtain well-posedness.


Let ν>0, z∈_≥ν, α∈[0,1]. Then

    z^α≥ ( z)^α≥ν^α.




Let us prove the first inequality. Note that without loss of generality, we may assume that z=1. Let φ z ∈-π2π2.
Since ln∘cos is concave on -π2π2 (as (ln∘cos)' = -tan is decreasing) and (ln∘cos)(0) = 0, we obtain

    lncos(αφ) = lncos(αφ + (1-α)0) ≥αlncos(φ) + (1-α)lncos(0) = ln(cos(φ)^α),

and therefore cos(αφ)≥cos(φ)^α.
Since z=1 implies z = 1/cos(φ), we obtain

    z^α = cos(αφ)/(cosφ)^α≥ 1 = ( z)^α.

The second inequality follows from monotonicity of x↦ x^α.


Applying prop:pdfrac and noting that D is boundedly invertible we can reformulate  eq:FEmatlaw as 

    ν^-αD^-1T-_0u=0,

so that eq:FEmatlaw and eq:FEmomentum read

    (ν[        ρ        0;        0 ν^-αD^-1 ]-[  0   ; _0  0 ])[ v; T ]=[ f; 0 ].

A solution theory for the latter equation, thus, reads as follows,
where again vνu.

 Put HŁ(Ω)^d×Ł(Ω)_sym^d× d.
Then for all ν>0 the operator

    ν[        ρ        0;        0 ν^-αD^-1 ]-[  0   ; _0  0 ]

is densely defined and closable in (;H). The inverse of the
closure is continuous, causal and eventually independent of ν.



The proof rests on thm:Solution_theory_EE. Since [  0   ; _0  0 ] is skew-selfadjoint by prop:block_op_realinv<ref>, it suffices to confirm the
positive definiteness condition for the material law. For this let
z∈_≥ν and compute for x∈Ł(Ω)_sym^d× d,
using prop:pdfrac and prop:block_op_realinv<ref>,

    xzz^-αD^-1x=xz^1-αD^-1x≥ν^1-αxD^-1x≥ν^1-αc/D^2x^2.

This yields the assertion.




§ THE HEAT EQUATION WITH DELAY


Let Ω⊆^d be open. In this section we concentrate
on a generalisation of the heat equation discussed in the previous
chapter. Although we keep the heat-flux-balance in the sense that 

    ∂_tθ+ q=Q,

with q×Ω→^d being the heat flux and θ×Ω→
being the heat, we shall now modify Fourier's law to the extent that

    q=-aθ-bτ_-hθ

for some a,b∈(Ł( with a≥ c
for some c>0, and h>0. We shall again assume homogeneous Neumann
boundary conditions for q. Written in the now standard block
operator matrix form, this modified heat equation reads

    (ν[ 1 0; 0 0 ]+[            0            0;            0 (a+bτ_-h)^-1 ]+[  0 _0;     0 ])[ θ; q ]=[ Q; 0 ].

In order to actually justify the existence of the operator (a+bτ_-h)^-1
 as a bounded linear operator, we provide the following lemma.




  * There exists ν_0>0 such that for all
ν≥ν_0 the operator a+bτ_-h is continuously invertible
in (;Ł(Ω)^d).

  * For all 0<c'<c/a^2 there is ν_1≥ν_0
such that for all z∈_≥ν_1 we have (a+b^-zh)^-1≥ c'. 




Note that a is invertible with a^-1≤1/c and a^-1≥c/a^2 by prop:block_op_realinv<ref>.


  * By exa:material_law_revisited<ref>, for all ν>0 we obtain

    bτ_-h_()≤b_(Ł(Ω)^d)sup_t∈^-(it+ν)h=b_(Ł(Ω)^d)^-hν.

Thus, we find ν_0>0 such that for all ν≥ν_0 we
obtain bτ_-ha^-1_()≤1/cbτ_-h_()<1.
Thus, 

    a+bτ_-h=(1+bτ_-ha^-1)a

is continuously invertible by a Neumann series argument.

  * Let 0<c'<c/a^2. We choose ν_1≥ν_0 such that b^-zha^-1_(Ł(Ω)^d)≤min{1/2,ε}
for all z∈_≥ν_1, where 0<ε≤1/2c(c/a^2-c').
For z∈_≥ν_1 we compute

    (a+b^-zh)^-1    = a^-1(1+b^-zha^-1)^-1
     =(a^-1∑_k=0^∞(-b^-zha^-1)^k)
        =(a^-1+∑_k=1^∞a^-1(-b^-zha^-1)^k)
       ≥c/a^2-∑_k=1^∞a^-1(-b^-zha^-1)^k
       ≥c/a^2-∑_k=1^∞a^-1(-b^-zha^-1)^k
       ≥c/a^2-1/c∑_k=1^∞(-b^-zha^-1)^k
        =c/a^2-1/c(-b^-zha^-1)/1-(-b^-zha^-1)≥c/a^2-1/c2ε≥ c'.*



With this lemma we are in the position to provide the well-posedness
for the modified heat equation, as well.

 Let H=Ł(Ω)×Ł(Ω)^d. There exists ν_0>0 such that for
all ν≥ν_0 the operator

    ν[ 1 0; 0 0 ]+[            0            0;            0 (a+bτ_-h)^-1 ]+[  0 _0;     0 ]

is densely defined and closable with continuously invertible closure
in (;H). The inverse
of the closure is causal and eventually independent of ν. 



The proof rests on thm:Solution_theory_EE and lem:delayInv.




§ DUAL PHASE LAG HEAT CONDUCTIONDUAL PHASE LAG HEAT CONDUCTION


The last example is concerned with a different modification of Fourier's
law. The heat-flux balance

    ∂_tθ+ q=Q

is accompanied by the modified Fourier's law

    (1+s_q∂_t+1/2s_q^2∂_t^2)q=-(1+s_θ∂_t)θ,

where s_q∈,s_θ>0 are given numbers, which are called
`phases'. 

The modified Fourier's law in eq:dualFL is an attempt
to resolve the problem of inifinite propagation speed which stems
from a truncated Taylor series expansion of a model given by 

    τ_s_qq=-τ_s_θθ.

Note that it can be shown that such a model would even be ill-posed,
see <cit.>.


Let us turn back to the system eq:dualHB and eq:dualFL.
Notice, since s_θ>0, and due to a strictly positive real
part of the derivative in our functional analytic setting, we deduce
that (1+s_θν) is continuously invertible for ν≥ 0. Thus,
we obtain

    ν(ν^-1+s_q+1/2s_q^2ν)(1+s_θν)^-1q=-θ

The block operator matrix formulation of the dual phase lag heat conduction
model is thus

    (ν[                               1                               0;                               0 (ν^-1+s_q+1/2s_q^2ν)(1+s_θν)^-1 ]+[  0 _0;     0 ])[ θ; q ]=[ Q; 0 ].


 Let H=Ł(Ω)×Ł(Ω)^d. Assume s_q∈∖{0}, s_θ>0. Then there
exists ν_0>0 such that for all ν≥ν_0 the operator

    ν[                               1                               0;                               0 (ν^-1+s_q+1/2s_q^2ν)(1+s_θν)^-1 ]+[  0 _0;     0 ]

is densely defined and closable with continuously invertible closure
in (;H). The inverse
of the closure is causal and eventually independent of ν. 


The proof of thm:wp-duallag is again based on thm:Solution_theory_EE.
Thus, we shall only record
the decisive observation in the next result. For this, we define 

    M(z)z^-1+s_q+1/2s_q^2z/1+s_θz∈  (z∈∖{0,-1s_θ}).



Let s_q∈∖{0}, s_θ>0. Then there exists ν_0∈
and c>0 such that for all z∈_≥ν_0 we have

    zM(z)≥ c.




We put σs_qs_θ. Let z∈∖{0,-1s_θ}.
We compute

    zM(z)=1+s_qz+1/2s_q^2z^2/1+s_θz=1/2s_qzσ+σ(1-1/2σ)+1-σ(1-1/2σ)/1+s_θz

and therefore 

    zM(z)=1/2s_qσ z+σ(1-1/2σ)+(1-σ(1-1/2σ))(1+s_θ z)/1+s_θz^2.

By assumption 

    0<s_q^2/s_θ=s_qσ,

and since 

    (1-σ(1-1/2σ))(1+s_θ z)/1+s_θz^2→ 0

as z→∞, we obtain

    zM(z)≥1/2s_qσ z-δ

for some δ>0 and all z∈ with z large
enough.




§ COMMENTS


The equations of poro-elasticity have been proposed in <cit.>
and were mathematically studied in <cit.>.

Equations of fractional elasticity are discussed in <cit.>.
The well-posedness conditions stated here and in exer:fracEla can
be generalised as it is outlined in <cit.> to the case
where both C and D are non-negative, selfadjoint operators
so that C and D satisfy the conditions imposed on N_1 and
N_0 in prop:EuclidM0M1. We refrained from presenting
this argument here, as it seemed too technical for the time being.
Note however that the proof is neither fundamentally different nor
considerably less elementary. 

The heat equation with delay has also been studied in <cit.>
with an entirely different strategy; the dual phase lag models have
been dealt with in <cit.>.

Other ideas to rectify infinite propagation speed of the heat equation
can be found in <cit.>, where nonlinear models for heat
conduction are being discussed.

The visco-elastic equations discussed in exer:integro-diif are
studied with convolution operators more general than below in <cit.>;
see also <cit.>.



§ EXERCISES

tocsectionExercises


 


  * 
Prove prop:UAU.

  * Prove thm:wp-Poro.

  * Let Ω⊆^d be open, ν>0, f∈ H_ν^1(;Ł(Ω)^d) and g∈ H_ν^1(;Ł(Ω)).
With the help of thm:wp-Poro show that for large enough
ν>0 there exist a unique u∈(ν^2)∩(λν)∩( C_0)
and p∈(ν)∩(α^*)∩(_0k)
such that 

    νρνu-λνu- C_0u+α^*p     =f
    νc_0p+ανu-_0k p     =g.





Let Ω⊆^d be open, C,D∈(Ł(Ω)_^d× d),
D=D^*≥ c for some c>0 and α∈[12,1]. Show that
there exists ν_0>0 such that for all ν≥ν_0 the
system 

    νρ v- T     =f,
    
    T     =(C+Dν^α)_0u,

where v=νu, admits a unique solution (v,T)∈(;Ł(Ω)^d×Ł(Ω)_^d× d)
for all f∈ H_ν^1(;Ł(Ω)^d).


The following exercises are devoted to showing the well-posedness
of certain equations in visco-elasticityvisco-elasticity,
where the `viscous part' is modelled by convolution with certain integral
kernels. The proof of the required positive definiteness property
requires some preliminary results. We assume the reader to be equipped
with the basics from the theory of functions of one complex variable.

For U⊆ open, u U→ holomorphic,
define f_ uŨ→ by f_ u(x,y) u(x+ı y)
for (x,y)∈Ũ(x,y)∈^2x+ı y∈ U.
We put

    H_(U)f_ uu U→ holomorphic.



Let U⊆ be open.


  * 
Let f∈ H_(U). Show that f satisfies the mean
value property; that is, for all (x,y)∈Ũ and r>0
with (x,y)r⊆Ũ we have

    f(x,y)=1/2π∫_0^2πf(x+rcosθ,y+rsinθ)θ̣.


  *  Let U_ >0 and f∈ H_(U)∩ C(×_≥ 0). Moreover, assume that f(x,0)=0 for each x∈ and f(x,y)→ 0 as |(x,y)|→∞. Show that f=0 on ×_≥ 0.





In this exercise we show a version of Poisson's formula.Poisson's formula  Let U_>0 and f∈ H_(U)∩ C(×_≥ 0). 


  * Assume that f(·,0)∈ L_p() for some 1≤ p<∞. Show that _>0∋ z↦1/π∫_ z'+ı( z-x')/( z-x')^2+( z)^2f(x',0)x̣'
is holomorphic.

  * Assume that f(·,0)∈ L_∞(). Show that 1/π∫_y/(x-x')^2+y^2) f(x',0)x̣'→ f(x_0,0) as x→ x_0 and y→ 0+.

  * (Poisson's formula)
Assume that f(·,0)∈ L_p() for some 1≤ p<∞ and f(x,y)→ 0 as |(x,y)|→∞ in ×_≥ 0. Show that

    f(x,y)=1/π∫_y/(x-x')^2+y^2f(x',0)x̣'   ((x,y)∈×_>0).

Hint: Apply exer:mvp<ref>.




Let ν_0∈ and k∈ L_1,ν_0(;)
with k⊆_≥0. 

  * 
Show that for all (x,ν)∈×ν_0 we have

    (ℒk)(ı x+ν)=1/π∫_ν-ν_0/(x-x')^2+(ν-ν_0)^2(ℒk)(ı x'+ν_0)x̣'.

Hint: Approximate k by functions in (_≥ 0;) and use Poisson's formula (see exer:Poisson).

  * 
Assume there exists d≥0 such that for all x∈ 

    x(ℒk)(ı x+ν_0)≤ d.

Show that for all ν≥ν_0 and x∈ we have

    x(ℒk)(ı x+ν)≤4d.

Hint: Use the formula in <ref> and split the integral into positive
and negative part of ; use symmetry of (ℒk) under conjugation
due to the realness of k.




 Let Ω⊆^d be open, ν_0∈
and k∈ L_1,ν_0(;) with k⊆_≥0.
Assume there exists d≥0 such that 

    x(ℒk)(ı x+ν_0)≤ d   (x∈).

Show that there exists ν_1≥ν_0 such that for all ν≥ν_1 the operator

    ν[         1         0;         0 (1-k*)^-1 ]+[  0   ; _0  0 ]

is well-defined, densely defined and closable in (;H) with H=Ł(Ω)^d×Ł(Ω)_^d× d.
Further, show that its closure is continuously invertible, and that the corresponding inverse
is causal and eventually independent of ν.



 Let ν_0∈ and k∈ L_1,ν_0(;)
with k⊆_≥0.


  * 
Assume that k is absolutely continuous with k'∈ L_1,ν_0(;). Show that there exist ν_1≥ν_0
and d≥0 with

    x(ℒk)(ı x+ν_1)≤ d  (x∈).



  * 
Assume that k(t)≥0 for all t∈ and that k(t)≤ k(s), whenever
s≤ t. Show that there exist ν_1≥ν_0 

    x(ℒk)(ı x+ν_1)≤ 0  (x∈).

Hint: For part (b) use the explicit formula for (ℒk) as an integral and the periodicity of sin.

NB: The condition in <ref> is a standard assumption for convolution
kernels in the framework of visco-elastic equations; the condition
in <ref> is from <cit.>.




[heading=subbibliography]



CHAPTER: CAUSALITY AND A THEOREM OF PALEY AND WIENER


In this chapter we turn our focus back to causal operators. In Chapter <ref> we
 found out that material laws provide a class of causal and autonomous
bounded operators. In this chapter we will present another proof of
this fact, which rests on a result which characterises functions in Ł(;H)
with support contained in the non-negative reals; the celebrated Theorem of Paley and Wiener. With
the help of this theorem, which is interesting in its own right, the
proof of causality for material laws becomes very easy. At a first
glance it seems that holomorphy of a material law is
a rather strong assumption. In the second part of this chapter, however, we
shall see that in designing autonomous and causal solution operators,
there is no way of circumventing holomorphy.

In the following, let H be a Hilbert space, and we consider (_≥0;H) as the subspace of functions in (;H) vanishing on -∞0.




§ A THEOREM OF PALEY AND WIENER


We start with the following lemma, for which we need the notion of locally integrable functions.
We define

    L_1,(;H)    f∀  K⊆ compact: _K f ∈ L_1(;H)
        = f∀ φ∈(): φ f ∈ L_1(;H).


Let f∈ L_1,(;H). Then we have f∈ L_2(_≥0;H)
if and only if f∈⋂_ν>0(;H) with sup_ν>0f_(;H)<∞.
In the latter case we have that 

    f_Ł(_≥0;H)=lim_ν→0f_(;H)=sup_ν>0f_(;H).




Let f∈Ł(_≥0;H) and ν>0. Then we estimate 

    ∫_f(t)_H^2^-2ν tṭ    =∫__≥0f(t)_H^2^-2ν tṭ≤∫__≥0f(t)_H^2ṭ
    =f_Ł(_≥0;H)^2,

which proves that f∈(;H) with f_(;H)≤f_Ł(_≥0;H)
for each ν>0. Moreover, f_(;H)→f_Ł(_≥0;H)
as ν→0 by monotone convergence and since clearly f_(;H)≤f_μ(;H)
for 0<μ≤ν we obtain 

    f_Ł(_≥0;H)=lim_ν→0f_(;H)=sup_ν>0f_(;H).

Assume now that f∈⋂_ν>0(;H) with Csup_ν>0f_(;H)<∞. This inequality yields

    sup_ν∈0∞∫_-∞0f(t)^2^-2ν tṭ≤ C.
Hence, the monotone convergence theorem yields that g(t)lim_ν→∞f(t)^2^-2ν t for t∈-∞0 defines a function g∈ L_1-∞0. Thus, [g=∞] is a set of measure zero and thus [f=0]=-∞0∖ [g=∞] has full measure in -∞0 implying that f⊆_≥0.

Finally, from 

    sup_ν∈0∞∫_0∞f(t)^2^-2ν tṭ≤ C.

we infer again by the monotone convergence theorem that t↦lim_ν→ 0f(t)^2^-2ν t=f(t)^2 defines a function in L_1(0,∞), showing the remaining assertion.


For the proof of the Paley–Wiener theorem we need a suitable space
of holomorphic functions on the right half-plane, the so-called Hardy spaceHardy
space (_>ν;H), which we introduce in the following.


For ν∈ we define the Hardy space

    (_>ν;H)g_>ν→ Hg holomorphic, sup_ρ>ν_g(ı t+ρ)_H^2ṭ<∞

and equip it with the norm ·_(_>ν;H) defined by

    g_(_>ν;H)sup_ρ>ν(_g(ı t+ρ)_H^2ṭ)^1/2.



We motivate the Theorem of Paley–Wiener first. For this, let f∈(_≥0;H) and define its Laplace transformLaplace transform
as

    _>ν∋  z↦ℒf(z) 1/√(2π)∫_0^∞ f(t)^-z tṭ.
Note that ℒf(z)=ℒ_ zf( z) for all z∈_>ν due to the support constraint on f. Moreover, it is not difficult to see that the integral on the right-hand side of eq:Lapl exists as (t↦^-ρ tf(t))∈ L_1(_≥0;H)∩Ł(_≥0;H)
for all ρ>ν.
Hence, ℒf_>ν→ H is holomorphic (cf. exer:analytic_on_strip). Moreover, by
lem:pos_support

    sup_ρ>νℒf(ı·+ρ)_Ł(;H)    =sup_ρ>νℒ_ρf_Ł(;H)
     =sup_ρ>νf_ρ(;H)
     =sup_ρ>0^-ν·f_ρ(;H)
        =^-ν·f_Ł(;H) = f_(;H),

which proves that 

    ℒ(_≥0;H)    →(_>ν;H)
    
    f    ↦(z↦(ℒ_ zf)( z))

is well-defined and isometric. It turns out that ℒ is actually surjective, see cor:Lapalce_unitary below. The difficult surjectivity statement is contained in the following Theorem of Paley–WienerTheorem of Paley–Wiener, <cit.>.
We mainly follow the proof given in <cit.>.



Let g∈(_>0;H). Then there exists an f∈Ł(_≥0;H) such that

    ℒ_νf=g(ı·+ν)  (ν>0).




For ν>0 we set g_ν g(ı·+ν)∈Ł(;H)
and f_νℱ^∗g_ν∈Ł(;H). Moreover,
we set f^(·)f_1. We first prove that f∈⋂_ν>0(;H)
with sup_ν>0f_(;H)<∞. For doing so, let
a>0, ρ>0 and x∈. Applying Cauchy's integral theorem
to the function z↦^zxg(z) and the curve γ, as indicated in Figure <ref>, we obtain 


    0     = ı_-a^a^(ı t+1)xg(ı t+1)ṭ-_ρ^1^(ı a+κ)xg(ı a+κ)κ̣
        -ı_-a^a^(ı t+ρ)xg(ı t+ρ)ṭ+_ρ^1^(-ı a+κ)xg(-ı a+κ)κ̣.

Moreover, since 

    __ρ^1^(±ı a+κ)xg(±ı a+κ)κ̣_H^2ạ   ≤__ρ^1^(±ı a+κ)x^2κ̣_ρ^1 g(±ı a+κ)_H^2κ̣ạ
       ≤_ρ^1^2κ xκ̣_ρ^1 _g(±ı a+κ)_H^2 ạκ̣
       ≤∫_ρ^1^2κ xκ̣1-ρg_(_>0;H)^2 <∞,

we infer that (a↦_ρ^1^(±ı a+κ)xg(±ı a+κ)κ̣)∈Ł(;H)
and thus, we find a sequence (a_n)_n∈ℕ in _>0
such that a_n→∞ and 

    _ρ^1^(±ı a_n+κ)xg(±ı a_n+κ)κ̣→0

as n→∞. Hence, using eq:Cauchy with a replaced
by a_n and letting n tend to infinity, we derive that

    _-a_n^a_n^(ı t+1)xg(ı t+1)ṭ-_-a_n^a_n^(ı t+ρ)xg(ı t+ρ)ṭ→0  (n→∞).

Noting that for each μ>0 we have

    _-a_n^a_n^(ı t+μ)xg(ı t+μ)ṭ=√(2π)^μ xℱ^∗(_[-a_n,a_n]g_μ)(x)   (x∈)

and that _[-a_n,a_n]g_μ→ g_μ in Ł(;H)
as n→∞, we may choose a subsequence (again denoted by (a_n))
such that

    0=    lim_n→∞( _-a_n^a_n^(ı t+1)xg(ı t+1)ṭ-_-a_n^a_n^(ı t+ρ)xg(ı t+ρ)ṭ)
    
    =    lim_n→∞(√(2π)^xℱ^∗(_[-a_n,a_n]g_1)(x)-√(2π)^ρ xℱ^∗(1_[-a_n,a_n]g_ρ)(x))
    
    =    √(2π)(^xf_1(x)-^ρ xf_ρ(x))

for almost every x∈. Hence, f=^(·)f_1=exp(ρ)f_ρ
for each ρ>0 and thus, 

    _f(t)_H^2^-2ρ tṭ=_f_ρ(t)_H^2ṭ<∞

which shows f∈⋂_ρ>0ρ(;H) with 

    sup_ρ>0f_ρ(;H)=sup_ρ>0f_ρ_Ł(;H)=sup_ρ>0g_ρ_Ł(;H)=g_(ℂ_>0;H).

Thus, f∈Ł(_≥0;H) with f_Ł(_≥0;H)=g_(ℂ_>0;H)
by lem:pos_support. Moreover, 

    ℒ_νf=ℱexp(-ν)f=ℱexp(-ν m)exp(ν)f_ν=ℱf_ν=g_ν=g(ı·+ν)

for each ν>0, which shows the representation formula for g.


Summarising the results of thm:Paley-Wiener and the arguments carried out just before thm:Paley-Wiener, we obtain the following statement.


Let ν∈. Then the mapping 

    ℒ(_≥0;H)    →(_>ν;H)
    
    f    ↦(z↦(ℒ_ zf)( z))

is an isometric isomorphism. In particular, (_>ν;H) is
a Hilbert space.



We have argued already that  ℒ is well-defined and isometric. Thus, we show that ℒ is onto, next. For this, let g∈(_>ν;H) and define g̃(z) g(z+ν)
for z∈_>0. Then g̃∈(_>0;H) and
thus, thm:Paley-Wiener yields the existence of f̃∈Ł(_≥0;H)
with 

    g(ı·+ρ)=g̃(ı·+ρ-ν)=ℒ_ρ-νf̃=ℒ_ρ(^ν·f̃ )  (ρ>ν).

Hence, setting f^ν·f̃∈(_≥0;H),
we obtain ℒf=g.

We can now provide an alternative proof of thm:material law independent of nu
by proving causality with the help of the Theorem of Paley–Wiener.

Let M(M)⊆→ L(H) be
a material law. Then for ν>M we have M(ν)∈((;H)) and M(ν)
is causal and autonomous (see exer:causality_autonomous).



Let ν>M. Then M_≥ν→ L(H) is bounded and
holomorphic on _>ν. Hence, by unitary equivalence, M(ν)∈ L((;H)).
Moreover, M(ν) is autonomous by exer:causality_autonomous.
Thus, for causality it suffices to check that M(ν)f⊆_≥0
whenever f∈(_≥0;H). So let f∈(_≥0;H).
Then ℒf∈(_>ν;H) by cor:Lapalce_unitary
and since M is bounded and holomorphic on _>ν, we infer also
that

    (z↦ M(z)(ℒf)(z))∈(_>ν;H).

Again by cor:Lapalce_unitary there exists g∈(_≥0;H)
such that 

    ℒg(z)=M(z)(ℒf)(z)  (z∈_>ν).

Thus, in particular

    ℒ_ρg=M(ı+ρ)ℒ_ρf  (ρ>ν).

Since f,g∈(_≥0;H) we infer that ℒ_ρg→ℒ_νg
and ℒ_ρf→ℒ_νf in Ł(;H) as
ρ→ν by dominated convergence. Moreover, M(ı+ρ)→ M(ı+ν)
strongly on Ł(;H) as ρ→ν (cf. exer:strong_conv_material law).
Hence, we derive 

    ℒ_νg=M(ı+ν)ℒ_νf,

and thus, g=M(ν)f which shows the causality.




§ A REPRESENTATION RESULT


In this section we argue that our solution theory needs holomorphy
as a central property for the material law. There are two key properties for rendering T∈ L(ν_0(;H)) a material law operator. The first one is causalitycausal (i.e., _-∞a()T_-∞a()= _-∞a()T for all a∈) and, secondly, T needs to be autonomousautonomous (i.e., τ_h T =Tτ_h for all h∈ where τ_hf=f(· +h)). The main theorem of this
section reads as follows:

 Let ν_0∈ and let T∈ L(ν_0(;H))
be causal and autonomous. Then T|_ν_0∩ has a
unique extension T_ν∈ L((;H)) for each ν>ν_0
and there exists a unique M_>ν_0→ L(H) holomorphic and
bounded such that T_ν=M(ν) for each ν>ν_0.

















































We consider the following (shifted) variant of thm:representaion first.

Let T∈ L(Ł(;H)) be causal and autonomous.
Then there exists M_>0→ L(H) holomorphic and bounded such
that 

    (ℒTf)(z)=M(z)(ℒf)(z)  (f∈Ł(_≥0;H),z∈_>0).




For s >0 and x∈ H we define f_x,s_0sx and compute

    ℒf_x,s(z) = 1/√(2π)∫_0^s ^-z tx ṭ =1/√(2π)1-^-z s/zx   (z∈_>0).

We define M_>0→ L(H) via

    M(z)x√(2π)z/1-^-zℒTf_x,1(z),
which is well-defined since Tf_x,1⊆ [0,∞) (use causality of T); M(z)∈ L(H), since T is bounded. Also, M(·)x is evidently holomorphic for every x∈ H as a product of two holomorphic mappings and thus by exer:holomorphic, M is holomorphic itself. Next, we show that for all z∈_>0 and f∈Ł(_≥ 0;H), we have

    (ℒTf)(z)=M(z)(ℒf)(z).

By definition of M, the equality is true for f replaced by f_x,1, x∈ H. Next, observe that _aa+1/nxa≥ 0, n∈, x∈ H is dense in Ł(_≥0;H). Hence, for eq:hendrik, it suffices to show

    (ℒT_aa+1/nx)(z)=M(z)(ℒ_aa+1/nx)(z)
 for all a≥ 0, n∈, x∈ H, and z∈_>0. Next, using that T is autonomous in the situation of eq:hendrik2, we see (T_aa+1/nx)=(Tτ_-a_01/nx)=τ_-a(T_01/nx) and, by a straightforward computation, (ℒτ_-af)(z)=^-zaℒf(z) for all f∈Ł(_≥ 0;H). Thus,

    (ℒT_aa+1/nx)(z) = ^-za(ℒT_01/nx)(z),
which yields that it suffices to show eq:hendrik2 for a=0 only, that is, for f=f_x,1/n. Furthermore, we compute for n∈ and z∈_>0

    ℒTf_x,1(z)    = ∑_k=0^n-1 (ℒT_k/n(k+1)/nx)(z)
        =  ∑_k=0^n-1^-zk/n(ℒT_01/nx)(z) =  1-^-z/1-^-z/n(ℒTf_x,1/n)(z).

Thus, using eq:hendrik0 for s=1/n, we deduce from the definition of M,

    ℒTf_x,1/n(z)    = 1-^-z/n/√(2π)z√(2π)z/1-^-zℒTf_x,1(z) 
        =  1-^-z/n/√(2π)zM(z)x 
        = M(z) ℒf_x,1/n(z).

Hence, eq:hendrik holds for all f∈Ł(_≥0;H). It remains to show boundedness of M. For this, let z∈_>0 and x∈ H. Set f_0∞^-z^*x as well as c 2 z√(2π). Then

    ℒf(z) = 1/√(2π)∫_0^∞^-zt-z^*txṭ = x/c.

By virtue of eq:hendrik, we get ℒTf(z)=M(z)ℒf(z) and thus M(z)x=cℒTf(z). This leads to

    M(z)x   ≤c/√(2π)∫_0^∞^-ztTf(t)ṭ≤c/√(2π)_0∞^-z(·)_Ł()Tf_Ł()
       ≤c/√(2π)_0∞^-z(·)_Ł()^2T_(Ł(;H))x_H = T_(Ł(;H))x_H,

where we used that f_Ł(;H)= _0∞^-z(·)_Ł()x_H. Thus, M(z)≤T, which yields boundedness of M and the assertion of the theorem.















































































We can now prove our main result of this section.


We just prove the existence of a function M. The proof of its
uniqueness is left as exer:Uniqueness_material_law.

We first prove the assertion for ν_0=0. So, let T∈ L(Ł(;H)) be
causal and autonomous. According to thm:weiss we find
M_>0→ L(H) holomorphic and bounded such that 

    (ℒTf)(z)=M(z)(ℒf)(z)  (f∈Ł(_≥0;H),z∈_>0).

Let now φ∈(;H) and set ainfφ.
Then τ_aφ∈Ł(_≥0;H), and for ν>0 we compute

    ℒ_νTφ    =ℒ_ντ_-aTτ_aφ
     =^-(ı+ν)aℒ_νTτ_aφ
     =^-(ı+ν)aM(ı+ν)ℒ_ντ_aφ
        =M(ı+ν)ℒ_νφ.

 The latter implies 

    Tφ_(;H)    =ℒ_νTφ_Ł(;H)
     =M(ı+ν)ℒ_νφ_Ł(;H)≤M_∞,_>0φ_(;H)

and hence, T|_(;H) has a unique continuous extension
T_ν∈ L((;H)). Using eq:unitar_equiv_on_dense_set
we obtain 

    T_ν=ℒ_ν^∗M(ı+ν)ℒ_ν=M(ν)

by approximation.

Let now ν_0∈. Then the operator 

    T̃^-ν_0T^ν_0∈ L(Ł(;H))

is causal and autonomous as well. Thus, T̃|_(;H)
has continuous extensions T̃_ρ∈ L(ρ(;H))
for each ρ>0 and there is M̃_>0→ L(H) holomorphic
and bounded such that T̃_ρ=M̃(ρ) for
each ρ>0. Using T|_(;H)=^ν_0T̃|_(;H)^-ν_0,
we derive that T|_(;H) has the unique continuous
extension T_ν=^ν_0T̃_ν-ν_0^-ν_0∈ L((;H))
for each ν>ν_0 and 

    ℒ_νT_ν    =ℒ_ν^ν_0T̃_ν-ν_0^-ν_0
     =ℒ_ν-ν_0T̃_ν-ν_0^-ν_0
     =M̃(ı+ν-ν_0)ℒ_ν-ν_0^-ν_0
        =M̃(ı+ν-ν_0)ℒ_ν.

Hence, 

    T_ν=M(ν)

for the holomorphic and bounded function M given by M(z)M̃(z-ν_0)
for z∈_>ν_0.




§ COMMENTS


The stated Theorem of Paley and Wiener is of course not the only theorem
characterising properties of the support of L_2-functions in
terms of their Fourier or Laplace transform. For instance, a similar
result holds for functions having compact support, see e.g. <cit.> and exer:PW_compact.
These theorems provide a nice connection between L_2-functions
and spaces of holomorphic functions in the form of the so-called Hardy spaces. In this
chapter we just introduced the Hardy space  and it is not surprising
that there are also the Hardy spaces ℋ_p for 1≤ p≤∞.
We refer to <cit.> for this topic.

The representation result presented in the second part of this chapter
was originally proved by Fours and Segal in 1955, <cit.>.
In this article the authors prove an analogous representation result
for causal operators on Ł(^d;H), where causality is defined
with respect to a closed and convex cone on ^d. The quite elementary
proof of thm:weiss for d=1 presented here was kindly communicated to us by Hendrik Vogt. 



§ EXERCISES

tocsectionExercises


Let Λ⊆_>0 be a set with
an accumulation point in _>0. Prove that {(x↦^-λ x) ; λ∈Λ}
is a total set in L_1(_≥0).

Hint: Use that the set is total if and only if 

    ∀ f∈ L_∞(_≥0): (∀λ∈Λ: __≥0^-λ xf(x)x̣=0⇒ f=0).








 Let M(M)⊆→ L(H)
be a material law. Moreover, let ν>M. Show that lim_ρ→νM(ı+ρ)=M(ı+ν)
where the limit is meant in the strong operator topology on Ł(;H). 









 Prove the uniqueness statement
in thm:representaion.



 Give an example of a continuous and bounded
function M_>0→ L(H) such that the corresponding operator
M(ν) is not causal for any ν>0.



 Prove the following distributional
variant of the Paley–Wiener theorem:
Let ν_0>0, k∈, f_>ν_0→, and set h(z)1/z^kf(z)
for z∈_>ν_0. We assume that h∈(_>ν_0;).
For ν>ν_0 we define the distribution u()→
by

    u(ψ)ℒ_ν^∗h(ı·+ν)(ν^∗)^kψ_(;)  (ψ∈(;)).

Prove that u⊆_≥0, where 

    u∖⋃U⊆ open∀ ψ∈(U;): u(ψ)=0.


What is u if f=__>ν_0?



 
 Let g∈Ł(),a>0 such that g⊆-aa. Show that fℱg extends to a holomorphic function f̃→ with f̃(ı t)=f(t) for each t∈ such that
 
    ∃ C≥ 0 ∀ z∈:  |f(z)|≤ C^a| z|.





 
 Let f:→ be holomorphic such that
 
  
  *  ∃ C≥ 0,  a>0 ∀ z∈:  |f(z)|≤ C^a| z|,
  
  *  f(ı·)∈Ł().
 
Prove that gℱ^∗ f(ı·) satisfies g⊆-aa.

Hint: Apply thm:Paley-Wiener to the function h:_>0→ given by 

    h(z)^-z af(z)/z+1   (z∈_>0)

to derive that g⊆_≥ -a.

Note: The assertion even holds true if one replaces condition <ref> by

    ∃ C≥ 0,  a>0 ∀ z∈:  |f(z)|≤ C^a|z|.








[heading=subbibliography]



CHAPTER: INITIAL VALUE PROBLEMS AND EXTRAPOLATION SPACES


Up until now we have dealt with evolutionary equations of the form

    (νM(ν)+A)U=F

for some given F∈(;H) for some Hilbert space H, a skew-selfadjoint
operator A in H and a material law M defined on a suitable
half space satisfying an appropriate positive definiteness condition with
ν∈ chosen suitably large. Under these conditions, we
 established that the solution operator, S_ν(νM(ν)+A)^-1∈((;H)),
is eventually independent of ν and causal; that is, if F=0
on -∞a for some a∈, then so too is U. 

Solving for U∈(;H) for some non-negative ν penalises
U having support on 0. This might be interpreted as an
implicit initial condition at -∞. In this chapter,
we shall study how to obtain a solution for initial value problems with an
initial condition at 0, based on the solution theory developed
in the previous chapters.



§ WHAT ARE INITIAL VALUES?


This section is devoted to the motivation of the framework to follow in the
subsequent section. Let us consider the following, arguably easiest
but not entirely trivial, initial value problem: find a `causal' u→
such that for u_0∈ we have

    u'(t)=0     (t>0),
    
    u(0)=u_0.

First of all note that there is no condition for u on 0.
Since, there is no source term or right-hand side supported on 0,
causality would imply that u=0 on -∞0. Moreover,
u=c for some constant c∈ on 0∞. Thus, in
order to match with the initial condition, 

    u(t)=u_0_0∞(t)  (t∈).

Notice also that u is not continuous. Hence, by the Sobolev embedding
theorem (thm:Sobolev_emb), u∉⋃_ν>0(ν).

Let H be a Hilbert space, u_0∈ H. Define 

    δ_0u_0(;H)    →
    
    f    ↦u_0f(0)_H.

Then, for all ν∈_>0, δ_0u_0 extends to a continuous
linear functional on (ν). Re-using the notation for
this extension, for all f∈(ν) we have

    (δ_0u_0)(f)=-_0∞u_0(ν-2ν)f_(;H).




The equality eq:delta_ibp is obvious for f∈(;H) as it is a
direct consequence of the fundamental theorem of calculus (look at the right-hand side first). The continuity
of δ_0u_0 follows from the Cauchy–Schwarz inequality
applied to the right-hand side of eq:delta_ibp. Note that
_0∞u_0∈(;H).

Recall from cor:adjoint_time_derivative that 

    ν^*=-ν+2ν.

Hence, if we formally apply this formula to eq:delta_ibp,
we obtain

    ν_0∞u_0f=_0∞u_0ν^*f=(δ_0u_0)(f).

Therefore, in order to use the introduced time derivative operator
for the above initial value problem, we need to extend the time derivative
to a broader class of functions than just (ν). To utilise the adjoint operator in this way will be central to the construction to
follow. It will turn out that indeed

    ν_0∞u_0=δ_0u_0.

Moreover, we shall show below that 

    νu=δ_0u_0

considered on the full time-line  is one possible replacement
of the initial value problem eq:IVP_initial.



§ EXTRAPOLATING OPERATORS


Since we are dealing with functionals, let us recall the definition
of the dual space.
Throughout this section let H,H_0,H_1 be Hilbert spaces.


The space

    H'φ H→φ linear and bounded

is called the dual space of H. We equip H' with the linear
structure

    (λ⊙φ+ψ)(x)λ^∗φ(x)+ψ(x)  (λ∈,φ,ψ∈ H',x∈ H).



Note that H' is a Hilbert space itself, since by the Riesz representation
theorem for each φ∈ H' we find a unique element R_Hφ∈ H
such that 

    ∀ x∈ H: φ(x)=R_Hφx.

Due to the linear structure on H', the so induced mapping R_H H'→ H (which is one-to-one and onto)
becomes linear and 

    H'× H'∋(φ,ψ)↦R_HφR_Hψ

defines an inner product on H', which induces the usual norm on
functionals. 


From now on we will identify elements x∈ H with their representatives in H'; that is, we identify x with R_H^-1x.

Let C(C)⊆ H_0→ H_1 be linear,
densely defined and closed. We recall that in this case (C)
endowed with the graph inner product

    (u,v)↦uv_H_0+CuCv_H_1

becomes a Hilbert space. Clearly, (C)↪ H_0
is continuous with dense range. Moreover, we see that (C)∋ x↦ Cx∈ H_1
is continuous. We define 

    C^♢ H_1   →(C)'  H^-1(C),
    
     (C^♢ϕ)(x)   ϕCx_H_1  (ϕ∈ H_1,x∈(C)).

Note that C^♢ is related to the dual operator C' of C considered as a bounded operator from (C) to H_1 by

    C^♢ =C'R_H_1^-1.



 With the notions and definitions from this
section, the following statements hold:


  *  C^♢ is continuous and linear.


  *  C^*⊆ C^♢.


  *  (C^*)=(C^♢).


  *  C⊆(C^*)^♢ H_0→(C^*)'=H^-1(C^*).


  *  H_0≅ H_0'↪ H^-1(C) densely and continuously.




<ref> Let ϕ,ψ∈ H_1, λ∈. Then

    C^♢(λϕ+ψ)(x) = λ^*(C^♢ϕ)(x) + (C^♢ψ)(x) = (λ⊙ C^♢ϕ + C^♢ψ)(x)   (x∈(C)).

To show continuity, let ϕ∈ H_1 and x∈(C). Then

    ϕCx_H_1≤ϕ_H_1Cx_H_1≤ϕ_H_1x_(C).

Hence, C^♢=sup_ϕ∈ H_1,ϕ_H_1≤1C^♢ϕ_(C)'≤1.

<ref> Let ϕ∈(C^∗). Then we have for all x∈(C)

    (C^♢ϕ)(x)=ϕCx_H_1=C^*ϕx_H_0=(C^*ϕ)(x).

We obtain C^♢ϕ=C^*ϕ. (Note that a functional on
H_0 is uniquely determined by its values on (C).)

<ref> Using <ref>, we are left with showing (C^♢)⊆(C^*).
So, let ϕ∈(C^♢). Then for all x∈(C)
we have

    0=(C^♢ϕ)(x)=ϕCx_H_1,

which leads to ϕ∈(C^*) and ϕ∈(C^*).

<ref> is a direct consequence of <ref> applied to
C^*.

<ref> Since (C)↪ H_0 is dense and continuous,
we obtain that H_0'↪(C)' is so as well; cf. exer:dual_embedding.

We will also write C_-1(C^*)^♢ for the so-called extrapolated operatorextrapolated operator of C. Then (C^*)_-1=C^♢. We will
record the index -1 at the beginning, but in order to avoid too
much clutter in the notation we will drop this index again, bearing
in mind that C_-1⊇ C and (C^*)_-1⊇ C^*.


We have shown that for all ν∈ the operator ν is
densely defined and closed. Then for f∈() we have for all
ϕ∈()

    ((ν)_-1f)(ϕ)     =fν^*ϕ_
     =f(-ν+2ν)ϕ_
     =-∫_f(^-2ν·ϕ)'_.

Hence, (ν)_-1f acts as the `usual' distributional derivative
taking into account the exponential weight in the scalar product.

With this observation we deduce that for ν>0 we have

    (ν)_-1_0∞=ν_0∞=δ_0.

Hence, the initial value problem from the beginning reads: find u
such that

    (ν)_-1u=δ_0u_0.




Let Ω⊆^d be open. Consider _0 H_0^1(Ω)⊆Ł(Ω)→Ł(Ω)^d.
We compute _-1Ł(Ω)^d→ H^-1(Ω) with
H^-1(Ω) H_0^1(Ω)'. For q∈Ł(Ω)^d
we obtain for all ϕ∈ H_0^1(Ω)

    (_-1q)(ϕ)     =q^*ϕ_Ł(Ω)^d
     =-q_0ϕ_Ł(Ω)^d.

Also, with similar arguments, we see that

    (()_-1f)(q)=-f_0q_Ł(Ω)

for all f∈Ł(Ω) and q∈ H_0(,Ω).


We consider a case of particular interest within the framework
of evolutionary equations.

Let A(C)×(C^*)⊆ H_0× H_1→ H_0× H_1
be given by

    A[ ϕ; ψ ]=[   0 C^*;  -C   0 ][ ϕ; ψ ]=[ C^*ψ;  -Cϕ ].

Then A_-1 H_0× H_1→ H^-1(C)× H^-1(C^*)
acts as

    A_-1[ ϕ; ψ ]=[        0 (C^*)_-1;    -C_-1        0 ][ ϕ; ψ ]=[ (C^*)_-1ψ;    -C_-1ϕ ].



Next, we will look at the solution theory when carried over to
distributional right-hand sides. 

An immediate consequence of the introduction of extrapolated operators, however, is
that we are now in the position to omit the closure bar for the operator sum in an evolutionary
equation, which we will see in an abstract version in thm:ComputeClosureDistr and for evolutionary equations in thm:NoClosureEEDistr.
The main advantage is that we can calculate an operator sum much easier than the closure of it. 
The price we have to pay is that we have to work in a larger space H^-1 of an operator in (;H) rather than in the original Hilbert space (;H).
Put differently, this provides another notion of “solutions” for evolutionary equations.
For this, we need to introduce the set 

    (H)ϕ(ϕ)⊆ H→ϕ linear

of not necessearily everywhere defined linear functionals on H.
Any u∈ H is thus identified with an element in (H) via
ψ↦uψ_H. Note that we can add and scalarly multiply elements in (H) with respect to the same addition and multiplication
defined on H' and with their natural domains. As usual, we will use the
⊆-sign for extension/restriction of mappings.



Let A(A)⊆ H→ H,
B(B)⊆ H→ H be densely defined and closed such that A+B
is closable, and assume that there exists (T_n)_n∈
in (H) such that T_n→_H in the strong operator topology
with (T_n)⊆(B) and

    T_nA    ⊆ AT_n,    T_nB ⊆ BT_n for all n∈.

Then
 T_n^∗  A^∗⊆ A^∗ T_n^∗ and T_n^∗ B^∗⊆ B^∗ T_n^∗ for each n∈ and (T_n^∗)⊆(B^∗).
Moreover, for x,f∈ H the following conditions are equivalent:


 
  * x∈(A+B) and (A+B)x=f.
 
  * A_-1x+B_-1x⊆ f∈(H).




Let n∈.  Taking adjoints in the inclusion T_nA⊆ AT_n, we derive (AT_n)^∗⊆ (T_nA)^∗. By thm:adj-comp and rem:adj-comp we obtain
 
    T_n^∗ A^∗⊆T_n^∗ A^∗=(AT_n)^∗⊆ (T_nA)^∗=A^∗ T_n^∗.

The same argument shows the claim for B^∗. Moreover, since BT_n is a closed linear operator defined on the whole space H, it follows that BT_n∈ L(H) by the closed graph theorem. Hence, (BT_n)^∗ is bounded by lem:bdds and since (BT_n)^∗⊆ (T_nB)^∗=B^∗ T_n^∗, we derive that (B^∗ T_n^∗)=H, showing that (T_n^∗)⊆(B^∗).

We now prove the asserted equivalence.

(i)⇒(ii): By definition, there exists (x_n)_n
in (A)∩(B) such that x_n→ x in H and Ax_n+Bx_n→ f.
By continuity, we obtain A_-1x_n→ A_-1x and B_-1x_n→ B_-1x
in H^-1(A^*) and H^-1(B^*), respectively. Thus, we have

    (A_-1x+B_-1x)(y)=lim_n→∞(A_-1x_n+B_-1x_n)(y)   =lim_n→∞Ax_n+Bx_ny=fy,

for each y∈(A^∗)∩(B^∗), which shows the asserted inclusion.

(ii)⇒(i): For n∈ we put x_n T_nx.
Then x_n∈(B) and for all y∈(A^*)∩(B^*),
we obtain 

    T_nf-Bx_ny    =T_nfy-T_nxB^*y
     =fT_n^*y-xT_n^*B^∗y
       =fT_n^*y-xB^*T_n^*y= f(T_n^∗ y)-(B_-1x)(T_n^*y) = (A_-1x)(T_n^∗ y) 
       =xA^*T_n^*y
     =xT_n^*A^*y
     =x_nA^*y,

where we have used that T_n^∗ y∈(A^∗)∩(B^∗). Let now y∈(A^∗). Then T_k^∗ y ∈(A^∗)∩(B^∗) for each k∈ and thus, by what we have shown above

    T_k(T_nf-Bx_n)y=T_nf-Bx_nT_k^∗ y=x_nA^* T_k^∗ y=x_nT_k^∗ A^∗ y=T_kx_nA^∗ y

for each k∈. Letting k tend to infinity, we derive

    T_n f-Bx_ny=x_nA^∗ y.

Since this holds for each y∈(A^∗), this implies that x_n∈(A) and Ax_n+Bx_n=T_nf.
Letting n→∞, we deduce x_n→ x and Ax_n+Bx_n→ f;
that is, (i).
























Let T(T)⊆ H→ H be
densely defined and closed with 0∈ρ(T). Then T_-1 H→ H^-1(T^*)
is an isomorphsim. In particular, the norms (T_-1)^-1·_H
and ·_H^-1(T^*) are equivalent.



Note that since 0∈ρ(T) we obtain {0}=(T)=((T^*)^♢)=(T_-1), see prop:CdiaCadj<ref>.
Thus, T_-1 is one-to-one. Next, let f∈ H^-1(T^*). Since
0∈ρ(T), we obtain 0∈ρ(T^*) by exer:resolvent_adj, which implies that T^*·T^*·
defines an equivalent scalar product on (T^*). Thus, by the
Riesz representation theorem, we find ϕ∈(T^*) such that
for all ψ∈(T^*) we have

    f(ψ)=T^*ϕT^*ψ=((T^*)^♢(T^*ϕ))(ψ).

Hence, f∈((T^*)^♢)=(T_-1), thus proving
that T_-1 is onto.

The following  alternative description of H^-1(T^*) is content of exer:completion.


Let T(T)⊆ H→ H be densely defined and closed with
0∈ρ(T). Then 

    H^-1(T^*)≅(̃H̃,̃T̃^̃-̃1̃·̃_̃H̃)̃,

where ≅ means isomorphic as Banach spaces and (̃·̃)̃ denotes
the completion.



Let B∈(H). Assume that T(T)⊆ H→ H is
densely defined and closed with 0∈ρ(T) and T^-1B=BT^-1.
Then B admits a unique continuous extension B∈(H^-1(T^*)).



By prop:CdiaCadj<ref>, (B)=H is dense in H^-1(T^*).
Thus, it suffices to show that B H⊆ H^-1(T^*)→ H^-1(T^*)
is continuous. For this, let ϕ∈ H and compute for all q∈(T^*)

    (Bϕ)(q)    =Bϕq
     =Bϕ(T^*)^-1T^*q
     =T^-1BϕT^*q
     =BT^-1ϕT^*q
       ≤BT^-1ϕq_(T^*).

The statement now follows upon invoking lem:ComputingDualNorm.

The abstract notions and concepts just developed will be applied to
evolutionary equations next.



§ EVOLUTIONARY EQUATIONS IN DISTRIBUTION SPACES


In this section, we will specialise the results from the previous
section and provide an extension of the solution theory in (;H).
For this, and throughout this whole section, we let H be a Hilbert space, μ∈ and
M_>μ→(H) be a material law.
Furthermore, let ν>max{M,0} and A(A)⊆ H→ H be
skew-selfadjoint.  In order to keep track
of the Hilbert spaces involved, we shall put

    H_ν^1(;H)    (ν),
    
    H_ν^-1(;H)   (ν)'≅(ν^*)'.



 Let H be a Hilbert space. Let D(D)⊆ H→ H be densely defined and closed and B∈(H). Assume that DB is densely defined. Then for all ϕ∈ H,  (DB)_-1(ϕ) = (D_-1B)(ϕ) on (D^*). 


First of all, note that (DB)^*=B^*D^*, by thm:adj-comp. Next, let ϕ∈ H and x∈(D^*). Then

    ((DB)_-1ϕ)(x)     =ϕ(DB)^*x  =ϕB^*D^*x
        =ϕB^*D^*x =BϕD^*x  = (D_-1Bϕ)(x).



The first application of the theory developed in the previous section
reads as follows.

Let U,F∈(;H). Then the following
statements are equivalent:



  * U∈(νM(ν)+A) and (νM(ν)+A)U=F.

  * νM(ν)U+AU⊆ F where the left-hand
side is considered as an element of H_ν^-1(;H)∩(;H^-1(A))⊆((;H)).



Before we come to the proof, we state the following lemma, whose proof is left as exer:proof_Lemma.


 

 
  * Let B:(B)⊆ H→ H and C:(C)⊆ H→ H be densely defined closed linear operators. Moreover, let λ,μ∈ρ(C) be in the same connected component of ρ(C) and
 
    (μ-C)^-1B⊆ B(μ-C)^-1.

Then (λ-C)^-1B⊆ B(λ -C)^-1. 

  * For ν>0 we have (1+εν)^-1→ 1_(;H) and (1+εν^∗)^-1→ 1_(;H) strongly as ε→ 0.





For n∈ we set T_n (1+1/nν)^-1. By lem:resolvTD we obtain T_n,T_n^∗→ 1_(;H) strongly in (;H) as n→∞. Moreover, by Hille's theorem (see prop:interchange_integral) we have ν^-1A⊆ Aν^-1 and thus, T_nA⊆ AT_n for each n∈ by lem:resolvTD, which also yields T_n^∗ A⊆ AT_n^∗ for each n∈ by thm:ComputeClosureDistr.
The latter, together with the strong convergence of (T_n) and (T_n^∗), yields that T_n,T_n^∗→ 1_(;(A)) strongly in (;(A)) as n→∞.

Consider the inclusion

    (νM(ν))_-1U+A_-1U⊆ F.

We show next that eq:-1-1 is equivalent to (ii) by applying prop:DB to the case D=ν, B=M(ν). For this assume that eq:-1-1 holds.
By prop:DB, we deduce that 

    ((νM(ν))_-1U+A_-1U)|_(ν^*)∩(A)=    ((ν)_-1M(ν)U+A_-1U)|_(ν^*)∩(A)

Thus, eq:-1-1 implies (ii). 

On the other hand, assume that (ii) holds. Let ϕ∈((νM(ν))^*)∩(;(A)). Then, for n∈, ϕ_n T_n^∗ϕ→ϕ as n→∞ in (;(A)) and
 
    (νM(ν))^*ϕ_n=T_n^∗(νM(ν))^*ϕ→ (νM(ν))^*ϕ  (n→∞)
 in (;H). By (ii) we obtain

    ((ν)_-1M(ν)U+A_-1U)(ϕ_n)=F(ϕ_n).

Using prop:DB, we infer

    ((νM(ν))_-1U+A_-1U)(ϕ_n)=F(ϕ_n).
Letting n→∞, we deduce eq:-1-1.

We are now in the position to apply thm:ComputeClosureDistr from above to the
case (;H) being the Hilbert space, A the operator in (;H), B=νM(ν), and T_n=(1+1/nν)^-1. We need to establish the commutativity properties next. The relation T_n A⊆ AT_n was already shown above.
Next, we infer (T_n)⊆(ν)⊆ (B) and 

    T_nB⊆ BT_n

for all n∈ by using the Fourier–Laplace transformation, see also thm:spectral-repr-derivative.
The closability of A+B is implied by thm:Solution_theory_EE.




Assume now that there exists c>0 such that

    zM(z)≥ c   (z∈_≥ν).

We recall from thm:Solution_theory_EE that the operator
νM(ν)+A
is continuously invertible in (;H).


 The operator S_ν(νM(ν)+A)^-1∈((;H))
admits a continuous extension to L(H_ν^-1(;H)). 



We apply prop:ContDistributionalExtn to (;H) being the Hilbert space,
T=ν and B=S_ν. For this, it remains to prove that
T^-1S_ν=S_νT^-1. This however follows from the fact
that z↦ S(z)(zM(z)+A)^-1 is a material
law and S(ν) = S_ν.




§ INITIAL VALUE PROBLEMS FOR EVOLUTIONARY EQUATIONS


Let H be a Hilbert space, μ∈,
M_>μ→(H) a material law, ν>max{M,0} and A(A)⊆ H→ H
skew-selfadjoint. 
In this section we shall focus on the implementation of initial value
problems for evolutionary equations. A priori there is no explicit
initial condition implemented in the theory established in (;H).
Indeed, choosing ν>0 we have only an implicit exponential decay
condition at -∞. For initial values at 0, we would rather want to
solve the following type of equation. In the situation of the previous
section, for a given initial value U_0∈ H we seek to solve 
the initial value problem

    (νM(ν)+A)U=0     on 0∞,
    
    U(0)=U_0.

In this generality the initial value problem cannot be solved. Indeed,
for U∈(;H) evaluation at 0 is not well-defined.
A way to overcome this difficulty is to weaken the attainment of the
initial value. For this, we specialise to the case when

    M(ν)=M_0+ν^-1M_1

with M_0, M_1∈(H).

We start with the following lemma, which will also be useful in the next chapter.


 Let U_0∈(A), U∈(;H) such that M_0U-_0∞M_0U_0:→ H^-1(A) is continuous,
U⊆0∞ and 

    νM_0U+M_1U+AU=0    on 0∞,
    
    M_0U(0)=M_0U_0   in  H^-1(A),

where the first equality is meant in the sense that

    ∀φ∈ H_ν^1(;H)∩(;(A)), φ⊆0∞:  (νM_0U+M_1U+AU)(φ)=0.

Then U-_0∞U_0∈(νM_0+M_1+A) and 

    (νM_0+M_1+A)(U-_0∞U_0)=-(M_1+A)U_0_0∞.




 We apply thm:NoClosureEEDistr for showing the claim; that is, we show that

    ((νM_0+M_1)(U-_0∞U_0)+A(U-_0∞U_0))(ψ)=(-(M_1+A)U_0_0∞)(ψ)

for each ψ∈ H_ν^1(;H)∩(;(A)). Note that by continuity, it suffices to show the equality for ψ∈(;(A)). So, let ψ∈(;(A)) and for n∈ we  define the function φ_n∈ H_ν^1() by   

    φ_n(t)
                            0    if  t≤ 0,
    
                            nt    if  t∈01/n,
    
                            1    if  t ≥ 1/n.

Note that φ_n ψ∈  H_ν^1(;H)∩(;(A)) and (φ_nψ) ⊆0∞ for each n∈. Thus, we obtain

    ((νM_0+M_1+A)(U-_0∞U_0))(ψ)
       = ((νM_0+M_1+A)U)(ψ)- ((νM_0+M_1+A)(_0∞U_0))(ψ)
       = ((νM_0+M_1+A)U)(φ_n ψ)+ ((νM_0+M_1+A)U)((1-φ_n)ψ)-
          -((νM_0+M_1+A)(_0∞U_0))(ψ)
       = ((νM_0+M_1+A)U)((1-φ_n)ψ)- (δ_0 M_0 U_0)(ψ)-((M_1+A)(_0∞ U_0))(ψ)

for each n∈. Thus, the claim follows if we can show that 

    ((νM_0+M_1+A)U)((1-φ_n)ψ)- (δ_0 M_0 U_0)(ψ)→ 0   (n→∞).

For doing so, we first observe that for all n∈ we have

    (δ_0 M_0 U_0)(ψ)=(δ_0 M_0 U_0)((1-φ_n)ψ)=(ν M_0 _0∞ U_0)((1-φ_n)ψ),

since φ_n(0)=0. Moreover,

    ((M_1+A)U)((1-φ_n)ψ)=U(1-φ_n)(M_1^∗+A^∗)ψ_→ 0   (n→∞),

since 1-φ_n() →_-∞0() strongly in (;H) and U⊆0∞. Thus, it remains to show that

    (νM_0(U-_0∞U_0))((1-φ_n)ψ)→ 0   (n→∞).

We compute

    (νM_0(U-_0∞U_0))((1-φ_n)ψ) 
        = M_0(U-_0∞U_0)ν^∗((1-φ_n)ψ)_
        =M_0(U-_0∞U_0)n_01/nψ_
      -M_0(U-_0∞U_0)(1-φ_n) νψ_
         +2νM_0(U-_0∞U_0)(1-φ_n)ψ_.

Note that the last two terms on the right-hand side tend to 0 as n→∞ since, as above, 1-φ_n() →_-∞0() strongly in (;H) and U⊆0∞. For the first term, we observe that

    M_0(U-_0∞U_0)n_01/nψ_
       ≤ n∫_0^1/nM_0 (U(t)-U_0)ψ(t)_H^-2ν tṭ
       ≤ n∫_0^1/nM_0 (U(t)-U_0)_H^-1(A)ψ(t)_(A^∗)^-2ν tṭ→ 0   (n→∞),

by the fundamental theorem of calculus, since (M_0U)(t)→ M_0U_0 in H^-1(A) as t→ 0.


Assume now additionally that there exists c>0 such that

    zM_0+M_1≥ c   (z∈_≥ν).

Then we can actually prove a stronger result than in the previous lemma.


Let U_0∈(A), U∈(;H). Then
the following statements are equivalent:



  * M_0U-_0∞M_0U_0→ H^-1(A) is continuous,
U⊆0∞ and 

    νM_0U+M_1U+AU=0    on 0∞,
    
    M_0U(0)=M_0U_0   in  H^-1(A),

where the first equality is meant as in lem:ivp.

  * U-_0∞U_0∈ (νM_0+M_1+A), and (νM_0+M_1+A)(U-_0∞U_0)=-(M_1+A)U_0_0∞.

  * U=S_νδ_0M_0U_0, with S_ν∈(H_ν^-1(;H)) as in thm:extSolTh.

Moreover, in either case we have M_0U-_0∞M_0U_0∈ H_ν^1(;H^-1(A)).



(i)⇒(ii): This was shown in lem:ivp.

(ii)⇒(iii): We have that

    U-_0∞U_0 =-S_ν((M_1+A)_0∞U_0).

Applying ν^-1 to both sides of this equality we infer that

    ν^-1(U-_0∞U_0)   =-S_ν((M_1+A)ν^-1_0∞U_0)
       = -ν^-1_0∞U_0+S_ν(νM_0 ν^-1_0∞U_0),

which gives 

    ν^-1U=S_ν(νM_0 ν^-1_0∞U_0)=S_ν(M_0_0∞U_0).

Applying ν to both sides and taking into account thm:extSolTh, we derive the claim.

(iii)⇒(ii): We do the argument in the proof of (ii)⇒(iii) backwards. First, we apply ν^-1 to
 U=S_ν(δ_0 M_0 U_0), which yields
 
    ν^-1U     = ν^-1S_ν(δ_0 M_0 U_0) = S_ν(M_0 _0∞ U_0) = S_ν(νM_0 ν^-1_0∞U_0).

 Thus,
 
    ν^-1(U-_0∞U_0)
        = S_ν(νM_0 ν^-1_0∞U_0)-ν^-1_0∞U_0
       =-S_ν((M_1+A)ν^-1_0∞U_0).

 An application of ν yields the claim.

(ii),(iii)⇒(i): 
Since U=S_ν(δ_0 M_0 U_0), we derive that

    (νM_0+M_1+A)U⊆δ_0M_0U_0,

which in particular yields (νM_0+M_1+A)U=0 on 0∞. 
By (ii) we infer 

    U-_0∞U_0 =-S_ν((M_1+A)_0∞U_0),

which shows that (U-_0∞U_0)⊆0∞ due to causality and hence, U⊆0∞. It remains to show that M_0(U-_0∞U_0)∈ H_ν^1(;H^-1(A)), since this would imply the continuity of M_0(U-_0∞U_0) with values in H^-1(A) by thm:Sobolev_emb and thus, 

    M_0(U-_0∞U_0)(0)=M_0(U-_0∞U_0)(0)=0  in  H^-1(A)

since the function is supported on 0∞ only. We compute 

    M_0(U-_0∞U_0)    =-M_0 S_ν((M_1+A)_0∞U_0)
       = νM_0 S_ν(ν^-1(M_1+A)_0∞U_0)
       = ν^-1(M_1+A)_0∞U_0-(M_1+A)S_ν(ν^-1(M_1+A)_0∞U_0),

and since the right-hand side belongs to H_ν^1(;H^-1(A)), the assertion follows.











































































































By thm:extSolTh, we always have U=S_νδ_0M_0U_0∈ H_ν^-1(;H).
This then serves as our generalisation for the initial value problem
even if U_0∉(A). 


The upshot of thm:IVPs_PDEs(ii) is that, provided U_0∈(A),  we can reformulate
initial value problems with the help of our theory as evolutionary equations with -right-hand
sides. Thus, we do not need the detour to extrapolation spaces
for being able to solve the initial value problem eq:IVP_PDE
(with an adapted initial condition as in (i)) in this situation.

Also note that  it may seem that U does depend on the `full
information' of U_0 as it is indicated in (ii). In fact, U
only depends on the values of U_0 orthogonal to the kernel of
M_0 as it is seen in (iii). We conclude this chapter with two
examples; the first one is the heat equation, the second example considers
Maxwell's equations.

[Initial Value Problems for the Heat Equation]
We recall the setting for the heat equation outlined in thm:wp_heat.
This time, we will use homogeneous Dirichlet boundary conditions for the heat
distribution θ. Let Ω⊆^d be open and bounded,
a∈ L_∞(Ω)^d× d with a(x)≥ c>0 for
a.e. x∈Ω for some c>0. In this case, we have 

    M_0    =[ 1 0; 0 0 ],  
    M_1 =[    0    0;    0 a^-1 ],  
    A =[  0   ; _0  0 ].

For the unknown heat distribution, θ, we ask it to have the
initial value θ_0∈(_0). Let ν>0 and V∈(;Ł(Ω)×Ł(Ω)^d) be the unique solution of

    (νM_0+M_1+A)V     =-(M_1+A)_0∞[ θ_0;   0 ] =-_0∞[     0; _0θ_0 ].

Then (θ,q) U V+_0∞[ θ_0;   0 ]∈(;Ł(Ω)×Ł(Ω)^d) satisfies (ii) from thm:IVPs_PDEs. Hence, on 0∞
we have

    [    νθ; a^-1q ]+[   q; _0θ ]=0

and the initial value is attained in the sense that 

    (M_0(θ,q))(0)=[ θ(0);    0 ]=[ θ_0;   0 ]  in   H^-1(A)=H^-1(_0)× H^-1(),

which follows from prop:A_neg where we computed H^-1(A). Let us
have a closer look at the attainment of the initial value. As a particular
consequence of strong convergence in H^-1(_0), we obtain
 for all ϕ∈() 
 
    θ(t)ϕ→θ_0ϕ

as t→0. Since _0 is one-to-one and has closed range,
we see that  has dense and closed range. Hence 
is onto. This implies that for all ψ∈Ł(Ω) 

    θ(t)ψ→θ_0ψ  (t→0).


We deduce that the initial value is attained weakly. This might seem
a bit unsatisfactory, however, we shall see stronger assertions for more particular
cases in the next chapter. 

Next, we have a look at Maxwell's equations.

[Initial Value Problems for Maxwell's equations]
We briefly recall the situation of Maxwell's equations from thm:st_maxwell.
Let ε,μ,σΩ→^3×3 satisfy
the assumptions in thm:st_maxwell and let (E_0,H_0)∈(_0)×().
Let (Ê,Ĥ)∈(;Ł(Ω)^6) satisfy

    (ν[ ε 0; 0 μ ]+[ σ 0; 0 0 ]+[  0  -; _0  0 ])[ Ê; Ĥ ]
        =-([ σ 0; 0 0 ]+[  0  -; _0  0 ])_0∞[ E_0; H_0 ] =_0∞[ -σ E_0+ H_0;      -_0E_0 ].

Then, as we have argued for the heat equation,

    [ E; H ][ Ê; Ĥ ]+_0∞[ E_0; H_0 ]

satisfies a corresponding initial value problem. We note here
that although often the second component in the right-hand side is
set to 0, as there are `no magnetic monopoles', in the theory of evolutionary equations the
second component of the right-hand side does appear as an initial
value in disguise.





§ COMMENTS


There are many ways to define spaces generalising the action of an
operator to a bigger class of elements; both in a concrete setting
and in abstract situations; see e.g. <cit.>.
People have also taken into account simultaneous extrapolation
spaces for operators that commute, see e.g. <cit.>. 

These spaces are particularly useful for formulating initial value
problems as was exemplified above; see also the concluding chapter
of <cit.> for more insight. Yet there is more to it as one can in fact generalise the
equation under consideration or even force the attainment of the initial
value in a stronger sense. These issues, however, imply that either
the initial value is attained in a much weaker sense, or that there are
other structural assumptions needed to be imposed on the material law M (as well as on
the operator A). 

In fact, quite recently, it was established that a particular proper subclass of evolutionary equations can be put into the framework of  C_0-semigroups. The conditions required to allow for statements in this
direction are, on the other hand, rather hard to check in practice;
see <cit.>.   







§ EXERCISES

tocsectionExercises


Let H_0 be a Hilbert space, T∈(H_0). Compute H^-1(T) and H^-1(T^*).



 Let H_0,H_1 be Hilbert spaces such that H_0↪ H_1 is dense and continuous. Prove that H_1'↪ H_0' is dense and continuous as well.




Prove the following statement which generalises prop:ContDistributionalExtn from
above: Let H_0 be a Hilbert space, A∈(H_0). Assume that T(T)⊆ H_0→ H_0 is
densely defined and closed with 0∈ρ(T) and T^-1A=AT^-1+T^-1BT^-1
for some bounded B∈(H_0). Then A admits a unique continuous
extension, A∈(H^-1(T^*)).



Let H_0 be a Hilbert space, N(N)⊆ H_0→ H_0 be a normal operator;
that is, N is densely defined and closed and NN^*=N^*N.
Show that H^-1(N)≅ H^-1(N^*) and deduce H^-1(ν)≅ H^-1(ν^*).


 
Prove prop:extrapolation_completion.




Let H_0 be a Hilbert space, n∈ and T(T)⊆ H_0→ H_0 be a densely
defined, closed linear operator with 0∈ρ(T). We define H^n(T)(T^n)
and H^-n(T) H^-1(T^n). Show that
for all k∈ and ℓ∈ we have that H^k+ℓ(T)↪ H^ℓ(T)
continuously and densely. Also show that 𝒟⋂_n∈(T^n)
is dense in H^ℓ(T) and dense in H^-ℓ(T^∗) for all ℓ∈ and that T|_𝒟
can be continuously extended to a topological isomorphism H^ℓ(T)→ H^ℓ-1(T)  and to an isomorphism H^-ℓ+1(T^∗)→ H^-ℓ(T^∗) for each ℓ∈.




Prove lem:resolvTD. 

Hint: Prove a similar equality with ν^-1 formally replaced
by z∈∂rr⊆ and deduce the assertion
with the help of thm:spectral-repr-derivative.


















[heading=subbibliography]



CHAPTER: DIFFERENTIAL ALGEBRAIC EQUATIONS


Let H be a Hilbert space and ν∈.
We saw in the previous chapter how initial value problems can
be formulated within the framework of evolutionary equations. More
precisely, we have studied problems of the form 

    (νM_0+M_1+A)U=0     on 0∞,
    
    M_0U(0)=M_0U_0

for U_0∈ H, M_0,M_1∈(H) and A(A)⊆ H→ H skew-selfadjoint; that is, we
have considered material laws of the form 

    M(z) M_0+z^-1M_1  (z∈∖{0}).

Here, the initial value is attained in a weak sense as an
equality in the extrapolation space H^-1(A). The first line
is also meant in a weak sense since the left-hand side turned out
to be a functional in H_ν^-1(;H)∩(;H^-1(A)).
In thm:IVPs_PDEs it was shown that the latter problem
can be rewritten as 

    (νM_0+M_1+A)U=δ_0M_0U_0.

In this chapter we aim to inspect initial value problems a little
closer but in the particularly simple case when A=0. However, we want to impose the initial condition for U and not just 
M_0U. Thus, we want to deal with the problem 

    (νM_0+M_1)U=0     on 0∞,
    
    U(0)=U_0

for two bounded operators M_0,M_1 and an initial value U_0∈ H.
This class of differential equations is known as differential
algebraic equations since the operator M_0 is allowed to have
a non-trivial kernel. Thus, eq:DAE is a coupled problem
of a differential equation
(on ( M_0)^) and an algebraic equation (on M_0). We begin by treating these equations in
the finite-dimensional case; that is, H=^n and M_0,M_1∈^n× n
for some n∈.



§ THE FINITE-DIMENSIONAL CASE


Throughout this section let n∈ and M_0,M_1∈^n× n.

We define the spectrum, matrix pairspectrum of the
matrix pair (M_0,M_1) by 

    σ(M_0,M_1)z∈(zM_0+M_1)=0,

and the resolvent set of the matrix pairresolvent set, matrix pair (M_0,M_1)
by 

    ρ(M_0,M_1)∖σ(M_0,M_1).






  * It is immediate that σ(M_0,M_1) is closed since
the mapping z↦(zM_0+M_1) is continuous.

  * Note in particular that the spectrum (the set of eigenvalues) of
a matrix A corresponds in this setting to the spectrum of the matrix pair (1,-A).



In contrast to the case of the spectrum of one matrix, it may happen
that σ(M_0,M_1)= (for example we can choose M_0=0 and M_1 singular).
More precisely, we have the following result.


The set σ(M_0,M_1) is
either finite or equals the whole complex plane . If σ(M_0,M_1)
is finite then σ(M_0,M_1)≤ n.



The function z↦(zM_0+M_1) is a polynomial of order
less than or equal to n. If it is constantly zero, then σ(M_0,M_1)=
and otherwise σ(M_0,M_1)≤ n.



The matrix pair (M_0,M_1) is called regularregular, matrix pair
if σ(M_0,M_1).

The main problem in solving an initial value problem of the form eq:DAE
is that one cannot expect a solution for each initial value U_0∈^n
as the following simple example shows.

Let M_0=[ 1 1; 0 0 ],  M_1=[ 1 0; 0 1 ] and let U_0∈^2. We assume that there exists a solution
U_≥0→^2 satisfying eq:DAE; that is, 

    U_1'(t)+U_2'(t)+U_1(t)     =0  (t>0),
    
    U_2(t)     =0  (t>0),
    
    U(0)     =U_0.

The second and third equation yield that the second coordinate of U_0
has to be zero. Then, for U_0=(x,0)∈^2 the unique solution
of the above problem is given by 

    U(t)=(U_1(t),U_2(t))=(x^-t,0)    (t≥0).




We call an initial value U_0∈^n consistent consistent initial value for eq:DAE,
if there exists ν>0 and U∈ C(_≥0;^n)∩(_≥0;^n) such that eq:DAE holds. We denote the
set of all consistent initial values for eq:DAE by 

    (M_0,M_1)U_0∈^nU_0 consistent.



It is obvious that (M_0,M_1) is a subspace of ^n.
In particular, 0∈(M_0,M_1).


It is now our goal to determine the space (M_0,M_1). One
possibility for doing so uses the so-called quasi-Weierstra normal
form.

 Assume that (M_0,M_1)
is regular. Then there exist invertible matrices P,Q∈^n× n
such that 

    PM_0Q=[ 1 0; 0 N ],   PM_1Q=[ C 0; 0 1 ],

where C∈^k× k and N∈^(n-k)×(n-k) for some
k∈{0,…,n}. Moreover, the matrix N is nilpotent; that is, there exists ℓ∈ such that N^ℓ=0.



Since (M_0,M_1) is regular we find λ∈ such that
λ M_0+M_1 is invertible.
We set P_1(λ M_0+M_1)^-1 and obtain 

    M_0,1    P_1M_0=(λ M_0+M_1)^-1M_0,
    
    M_1,1    P_1M_1=(λ M_0+M_1)^-1M_1=1-λ M_0,1.

Let now P_2∈^n× n such that 

    M_0,2 P_2M_0,1P_2^-1=[ J 0; 0 Ñ ]

for some invertible matrix J∈^k× k and a nilpotent matrix
Ñ∈^(n-k)×(n-k) (use the Jordan normal form
of M_0,1 here). Then 

    M_1,2 P_2M_1,1P_2^-1=[ 1-λ J     0;     0  1-λÑ ].

Now, by the nilpotency of Ñ, the matrix (1-λÑ) is
invertible by the Neumann series. We set 

    P_3[      J^-1         0;         0 (1-λÑ)^-1 ]

and obtain 

    P_3M_0,2=[          1          0;          0 (1-λÑ)^-1Ñ ], and    P_3M_1,2=[ J^-1-λ      0;      0      1 ].

Note that (1-λÑ)^-1Ñ is nilpotent, since
the matrices commute and Ñ is nilpotent. Thus, the assertion
follows with N(1-λÑ)^-1Ñ, C J^-1-λ,
P=P_3P_2P_1, and Q=P_2^-1.


It is clear that the matrices P, Q, C and N in the previous proposition
are not uniquely determined by M_0 and M_1. However, the size of N and C
as well as the degree of nilpotency of N are determined by M_0
and M_1 as the following proposition shows.


Let P,Q∈^n× n be invertible
such that 

    PM_0Q=[ 1 0; 0 N ],   PM_1Q=[ C 0; 0 1 ],

where C∈^k× k, N∈^(n-k)×(n-k) for some
k∈{0,…,n}, and N is nilpotent. Then (M_0,M_1)
is regular and


  *  k is the degree of the polynomial z↦(zM_0+M_1).

  *  N^ℓ=0 if and only if

    sup_|z|≥ rz^-ℓ+1(zM_0+M_1)^-1<∞

for one (or equivalently all) r>0 such that 0r⊇σ(M_0,M_1).




First, note that

    (zM_0+M_1)=1/ P  Q[  z+C    0;    0 zN+1 ]=1/ P  Q(z+C)  (z∈).

Hence, (M_0,M_1) is regular and 

    k=((·)+C)=((·)M_0+M_1),

which shows <ref>. Moreover, we have ρ(M_0,M_1)=ρ(-C) and

    (zM_0+M_1)^-1=Q[  (z+C)^-1         0;         0 (zN+1)^-1 ]P  (z∈ρ(M_0,M_1)),

and hence, for r>0 with 0r⊇σ(M_0,M_1)
we have

    (zM_0+M_1)^-1≤ K_1(zN+1)^-1  (|z|≥ r)

for some K_1≥0, since sup_|z|≥ r(z+C)^-1<∞.
Now let ℓ∈ such that N^ℓ=0. Then

    (zN+1)^-1=∑_k=0^ℓ-1(-1)^kz^kN^k≤ K_2z^ℓ-1  (z≥ r)

for some constant K_2≥0 and thus, 

    (zM_0+M_1)^-1≤ K_1K_2z^ℓ-1  (|z|≥ r).

Assume on the other hand that 

    sup_|z|≥ rz^-ℓ+1(zM_0+M_1)^-1<∞

for some ℓ∈ and r>0 with σ(M_0,M_1)⊆0r.
Then there exist K̃_1,K̃_2≥0 such that

    (zN+1)^-1≤[  (z+C)^-1         0;         0 (zN+1)^-1 ]≤K̃_1(zM_0+M_1)^-1≤K̃_̃2̃z^ℓ-1

for all z∈ with z≥ r.
Now, let p∈ be minimal such that N^p=0. We show that p≤ℓ
by contradiction. Assume p>ℓ. Then we compute 

    0    =lim_n→∞1/n^ℓ(nN+1)^-1N^p-ℓ-1=lim_n→∞∑_k=0^p-1(-1)^kn^k-ℓN^k+p-ℓ-1
        = lim_n→∞∑_k=0^ℓ-1(-1)^kn^k-ℓN^k+p-ℓ-1 + (-1)^ℓN^p-1
        = (-1)^ℓN^p-1,

which contradicts the minimality of p.


Let (M_0,M_1) be regular
and P,Q∈^n× n be chosen according to prop:Weierstrass_normal_form. Let k=((·)M_0+M_1).
Then 

    (M_0,M_1)=U_0∈^nQ^-1U_0∈^k×{0}.

Moreover, for each U_0∈(M_0,M_1)
the solution U of eq:DAE is unique and satisfies U∈ C(_≥0;^n)∩ C^1(_>0;^n)
as well as 

    M_0U'(t)+M_1U(t)     =0  (t>0),
    
    U(0)     =U_0.




Let C∈^k× k and N∈^(n-k)×(n-k) be nilpotent
as in prop:Weierstrass_normal_form. Obviously U is
a solution of eq:DAE if and only if V Q^-1U
is continuous on _≥0 and solves 

    (ν[ 1 0; 0 N ]+[ C 0; 0 1 ])V     =0  on 0∞,
    
    V(0)     =Q^-1U_0 V_0.

Clearly, if Q^-1U_0=(x,0)∈^k×{0} then V given by V(t)(^-tCx,0) for t≥0 is a solution of eq:auxP for ν>0 large enough.
On the other hand, if V given by V(t)=(V_1(t),V_2(t))∈^k×^n-k (t≥ 0) is
a solution of eq:auxP then we have 

    νNV_2+V_2=0  on 0∞.

Since N is nilpotent, there exists ℓ∈ with N^ℓ=0.
Hence, 

    N^ℓ-1V_2(t) = -N^ℓ-1νNV_2(t) = νN^ℓ V_2(t) =0  (t>0),

which in turn implies νN^ℓ-1V_2=0 on 0∞.
Using again the differential equation, we infer N^ℓ-2V_2(t)=0
for t>0. Inductively, we deduce V_2(t)=0 for t>0
and by continuity V_2(0)=0, which yields V_0=Q^-1U_0∈^k×{0}.
The uniqueness follows from prop:uniqueness_DAE below.




§ THE INFINITE-DIMENSIONAL CASE


Let now M_0,M_1∈ L(H).
Again, it is our aim to determine the space of consistent initial
values for the problem 

    (νM_0+M_1)U=0    on 0∞,
    
    U(0)=U_0.


Here, consistent initial values are defined as in the finite-dimensional
setting:

We call an initial value U_0∈ H consistent consistent initial value for eq:DAE_infinite,
if there exist ν>0 and U∈ C(_≥0;H)∩(_≥0;H) such that eq:DAE_infinite holds. We denote
the set of all consistent initial values for eq:DAE_infinite by 

    (M_0,M_1)U_0∈ HU_0 consistent.


Before we try to determine (M_0,M_1) we prove a regularity
result for solutions of eq:DAE_infinite.

Let ν>0, U_0∈ H and U∈ C(_≥0;H)∩(_≥0;H)
be a solution of eq:DAE_infinite.
Then M_0(U-_0∞U_0)∈ H_ν^1(;H) and 

    νM_0(U-_0∞U_0)+M_1U=0.




  We extend U to  by 0.
  First, observe that M_0 (U - _0∞ U_0)→ H is continuous, since U is continuous and U(0) = U_0.
  By lem:ivp (with A=0), we obtain U-_0∞ U_0∈(νM_0 + M_1) and (νM_0+M_1)(U - _0∞ U_0) = -M_1U_0 _0∞. Since ν is closed and M_0 is bounded, νM_0 is closed as well. Since M_1 is bounded, therefore also
  νM_0+M_1 is closed. Thus,
  U-_0∞U_0∈(νM_0 + M_1) = (ν M_0) and therefore M_0(U-_0∞U_0)∈(ν), and
  
    νM_0(U - _0∞ U_0) + M_1U = 0.



We now come back to the space (M_0,M_1). Since we
are now dealing with an infinite-dimensional setting, we cannot use
normal forms to determine (M_0,M_1) without dramatically restricting
the class of operators. Thus, we follow a different approach
 using so-called Wong sequences.

We set 

    _0 H

and for k∈_0 we set 

    _k+1 M_1^-1[M_0[_k]].

The sequence (_k)_k∈_0 is called the Wong sequenceWong sequence
associated with (M_0,M_1).


By induction, we infer _k+1⊆_k for each k∈_0.


As in the matrix case, we denote by 

    ρ(M_0,M_1)z∈(zM_0+M_1)^-1∈(H)

the resolvent set of (M_0,M_1).

Let k∈_0. Then:


  *  M_1(zM_0+M_1)^-1M_0=M_0(zM_0+M_1)^-1M_1
for each z∈ρ(M_0,M_1).

  *  (zM_0+M_1)^-1M_0[_k]⊆_k+1 for
each z∈ρ(M_0,M_1).

  *  If x∈_k we find x_1,…,x_k+1∈ H such
that for each z∈ρ(M_0,M_1)∖{0}

    (zM_0+M_1)^-1M_0x=1/zx+∑_ℓ=1^k1/z^ℓ+1x_ℓ+1/z^k+1(zM_0+M_1)^-1x_k+1.


  *  If ρ(M_0,M_1)∅ then M_1^-1[M_0[_k]]⊆_k+1.




The proof of the statements <ref> to <ref> are left as exer:lemma_wong_sequence. We now prove <ref>. If k=0 there is nothing to show. So assume that
the statement holds for some k∈_0 and let x∈ M_1^-1[M_0[_k+1]].
Since _k+1⊆_k, we infer
x∈ M_1^-1[M_0[_k]]⊆_k+1
by induction hypothesis. Hence, we find a sequence (w_n)_n∈
in _k+1 with w_n→ x. Let now z∈ρ(M_0,M_1).
Then, by <ref>, we have (zM_0+M_1)^-1M_0w_n⊆_k+2 for
each n∈ and hence, (zM_0+M_1)^-1M_0x∈_k+2.
Moreover, since M_1x∈ M_0[_k+1],
we find a sequence (y_n)_n∈ in _k+1 with M_0y_n→ M_1x.
Setting now 

    x_n(zM_0+M_1)^-1zM_0x+(zM_0+M_1)^-1M_0y_n∈_k+2

(where, again, we have used <ref>) for n∈, we derive 

    x_n    =(zM_0+M_1)^-1zM_0x+(zM_0+M_1)^-1M_0y_n
     =x-(zM_0+M_1)^-1(M_1x-M_0y_n) 
       → x

as n→∞ and thus, x∈_k+2.


The importance of the Wong sequence becomes apparent if we consider
solutions of eq:DAE_infinite.


Assume that ρ(M_0,M_1)∅.
Let ν>0 and U∈(_≥0;H)∩ C(_≥0;H)
be a solution of eq:DAE_infinite. Then U(t)∈⋂_k∈_0_k
for each t≥0.



We prove the claim, U(t)∈_k for all t≥ 0 and k∈_0, by induction. For k=0 there is nothing to show.
Assume now that U(t)∈_k for each t≥0 and
some k∈_0. By prop:regularity_DAE we know that 

    νM_0(U-_0∞U_0)+M_1U=0

and thus, in particular, 

    M_0U(t)-M_0U_0+∫_0^tM_1U(s)ṣ=0  (t≥0).

Let now t≥0 and h>0. Then we infer 

    M_0U(t+h)-M_0U(t)+M_1∫_t^t+hU(s)ṣ=0

and hence, 

    ∫_t^t+hU(s)ṣ∈ M_1^-1[M_0[_k]]⊆_k+1

by lem:properties-Wong<ref>. Since U is continuous,
the fundamental theorem of calculus implies U(t)∈_k+1,
which yields the assertion.


In particular, the space of consistent initial values has to be a
subspace of ⋂_k∈_0_k. We now impose an
additional constraint on the operator pair (M_0,M_1), which is equivalent to being regular
in the finite-dimensional setting (cf. prop:index_and_uniqueness).


We call the operator pair (M_0,M_1) regular if there
exists ν_0≥0 such that


  * _>ν_0⊆ρ(M_0,M_1), and

  *  there exist C≥0 and ℓ∈ such that for all z∈_>ν_0 we have (zM_0+M_1)^-1≤ Cz^ℓ-1.

Moreover, we call the smallest ℓ∈ satisfying <ref> the index
of (M_0,M_1)index of operator pair, which is denoted by (M_0,M_1).



Note that for matrices M_0 and M_1 the index equals the
degree of nilpotency of N in the quasi-Weierstra normal form by prop:index_and_uniqueness.


From now on, we will require that (M_0,M_1) is regular. First, we
prove an important result on the Wong sequence in this case.

 Let (M_0,M_1) be regular, k∈_0, and k≥(M_0,M_1). Then

    _k=_(M_0,M_1).




We show that _k=_k+1 for each k≥(M_0,M_1).
Since the inclusion “⊇” holds trivially, it suffices
to show _k⊆_k+1. For doing so, let
k≥(M_0,M_1) and x∈_k. By lem:properties-Wong<ref> we find x_1,…,x_k+1∈ H such that 

    (zM_0+M_1)^-1M_0x=1/zx+∑_ℓ=1^k1/z^ℓ+1x_ℓ+1/z^k+1(zM_0+M_1)^-1x_k+1

for each z∈_>ν_0. Since k≥(M_0,M_1),
we derive 

    z(zM_0+M_1)^-1M_0x→ x  ( z→∞),

and since the elements on the left-hand side belong to _k+1,
by lem:properties-Wong<ref>, the assertion immediately follows.


We now prove that in case of a regular operator pair (M_0,M_1)
the solution of eq:DAE_infinite for a consistent initial
value U_0 is uniquely determined.


 Let (M_0,M_1) be regular, U_0∈(M_0,M_1), and ν>0
such that a solution U∈ C(_≥0;H)∩(_≥0;H) of eq:DAE_infinite exists. Then this solution is unique.


In particular

    (ℒ_ρU)(t)=1/√(2π)((ı t+ρ)M_0+M_1)^-1M_0U_0  (a.e. t∈)

for each ρ>max{ν,ν_0}. 



By prop:regularity_DAE we have M_0(U-_0∞U_0)∈ H_ν^1(;H)
and 

    νM_0(U-_0∞U_0)+M_1U=0.

Applying the Fourier–Laplace transformation, ℒ_ρ, for ρ>max{ν,ν_0}
the latter yields 

    (ı t+ρ)M_0(ℒ_ρU(t)-1/√(2π)1/ı t+ρU_0)+M_1ℒ_ρU(t)=0  (a.e. t∈)

which in turn yields 

    ℒ_ρU(t)=1/√(2π)((ı t+ρ)M_0+M_1)^-1M_0U_0  (a.e. t∈)

and, in particular, proves the uniqueness of the solution. 



The formula in prop:uniqueness_DAE shows that U∈ν_0(;H)
for all solutions U of eq:DAE_infinite for a consistent
initial value U_0 and hence, we also have M_0(U-_0∞U_0)∈ H_ν_0^1(;H).


One interesting consequence of the latter proposition is the following.

Let (M_0,M_1) be regular. Then the
operator M_0(M_0,M_1)→ H is injective.



Let U_0∈(M_0,M_1) with M_0U_0=0. By prop:uniqueness_DAE,
the solution U of eq:DAE_infinite with U(0)=U_0
satisfies 

    ℒ_ρU(t)=((ı t+ρ)M_0+M_1)^-1M_0U_0=0

and hence, U=0, which in turn implies U_0=U(0)=0.


We now want to determine the space (M_0,M_1) in terms of
the Wong sequence.

 Let (M_0,M_1) be regular. Then

    _(M_0,M_1)⊆(M_0,M_1)⊆_(M_0,M_1).




The second inclusion follows from lem:initial_values_Wong-sequence
and prop:Wong-terminates. Let now U_0∈_(M_0,M_1)
and set 

    V(z)1/√(2π)(zM_0+M_1)^-1M_0U_0  (z∈_>ν_0).

Let k(M_0,M_1). By lem:properties-Wong<ref> we find x_1,…,x_k+1∈ H such that 

    V(z)=1/√(2π)(1/zU_0+∑_ℓ=1^k1/z^ℓ+1x_ℓ+1/z^k+1(zM_0+M_1)^-1x_k+1)  (z∈_>ν_0).

In particular, we read off that V∈(_>ν_0;H) and
hence, by the Theorem of Paley–Wiener (more precisely by cor:Lapalce_unitary)
there exists U∈ν_0(_≥0;H) such that 

    (ℒ_ρU)(t)=V(ı t+ρ)  (a.e. t∈,ρ>ν_0).

Moreover, 

    zV(z)-1/√(2π)U_0=1/√(2π)(∑_ℓ=1^k1/z^ℓx_ℓ+1/z^k(zM_0+M_1)^-1x_k+1)  (z∈_>ν_0)

and hence (z↦ zV(z)-1/√(2π)U_0)∈(_>ν_0;H) as well.
Since 

    (ℒ_ρρ(U-_0∞U_0))(t)     =(ı t+ρ)(ℒ_ρU)(t)-1/√(2π)U_0
        =(ı t+ρ)V(ı t+ρ)-1/√(2π)U_0  (a.e. t∈,ρ>ν_0),

we infer U-_0∞U_0∈ H_ν_0^1(;H) and,
thus, U-_0∞U_0 is continuous by thm:Sobolev_emb.
Hence, U∈ C(_≥0;H) and since U⊆_≥0
we derive U(0)=U_0. Finally, by the definition of V,

    M_0(zV(z)-1/√(2π)U_0)=-1/√(2π)M_1(zM_0+M_1)^-1M_0U_0=-M_1V(z)  (z∈_>ν_0).

Hence, 

    ν_0M_0(U-_0∞U_0)+M_1U=0,

from which we see that U solves eq:DAE_infinite.


Finally, we treat the case when (M_0,M_1) is closed. 



Let (M_0,M_1) be regular and (M_0,M_1) closed. Then the
operator S(M_0,M_1)→ C(_≥0;H), which assigns to each
initial state, U_0∈(M_0,M_1), its corresponding solution, U∈ C(_≥0;H), of eq:DAE_infinite is bounded in the
sense that 

    S_n(M_0,M_1)→ C([0,n];H),   U_0↦ SU_0|_[0,n]

is bounded for each n∈.



By prop:IV_k-in-IV we infer that (M_0,M_1)=_k
with k(M_0,M_1). Let ν>ν_0≥0. By prop:uniqueness_DAE and cor:Lapalce_unitary,
there exists C≥0 such that

    ν^-kSu_0_ν(0;H)=(z↦ z^-k(zM_0+M_1)^-1M_0U_0)_(_>ν;H)≤ C√(π/ν)M_0U_0_H

for each U_0∈(M_0,M_1), where
we have used the regularity of (M_0,M_1) and 

    (z↦ z^-1M_0U_0)_(_>ν;H)=√(π/ν)M_0U_0_H.

In particular, S(M_0,M_1)→ H^-1(ν^k) is bounded. Since ν_0(_≥0;H)↪ H^-1(ν^k)
continuously, we infer that S(M_0,M_1)→ν_0(_≥0;H)
is bounded by the closed graph theorem. Hence, also 

    S_n(M_0,M_1)→Ł([0,n];H),   U_0↦ SU_0|_[0,n]

is bounded for each n∈ and since C([0,n];H)↪Ł([0,n];H)
continuously, we infer that S_n is bounded with values in C([0,n];H)
again by the closed graph theorem.



 The variant of the closed graph theorem used in the proof above is the following: Let X,Y be Banach spaces and Z a Hausdorff topological vector space (e.g. a Banach space) such that Y↪ Z continuously. Let T X→ Z be linear and continuous with T[X]⊆ Y. Then T∈(X,Y). Indeed, by the closed graph theorem it suffices to show that T:X→ Y is closed. For doing so, let (x_n)_n be a sequence in X with x_n→ x and Tx_n → y for some x∈ X, y∈ Y. Then Tx_n→ Tx in Z by the continuity of T and Tx_n→ y in Z be the continuous embedding. Hence, y=Tx and thus, T is closed.





§ COMMENTS


The theory of differential algebraic equations in finite dimensions is a very
active field. The main motivation for studying these equations comes
from the modelling of electrical circuits and from control theory
(see e.g. <cit.> and exer:concrete_example). The main reference for the statements
presented in the first part of this lecture is the book by Kunkel
and Mehrmann <cit.>. Of course, also in the finite-dimensional
case Wong sequences can be used to determine the consistent initial
values, see exer:Wong-sequence-matrix. For instance, in <cit.> the connection between
Wong sequences and the quasi-Weierstra normal form for matrix pairs
is studied. Of course, the theory is not restricted to linear and
homogeneous problems. Indeed, in the non-homogeneous case it turns
out that the set of consistent initial values also depends on the
given right-hand side.

The theory of differential algebraic equations in infinite dimensions is less well studied than the finite-dimensional case.
We refer to <cit.>, where the theory of C_0-semigroups
is used to deal with such equations. Moreover, we refer to <cit.>, where sequences of projectors are used to decouple the system. Moreover, there exist several references in the Russian literature, where the equations are called Sobolev type equations (see e.g. <cit.>). The results on infinite-dimensional
problems presented here are based on <cit.>. In <cit.> the focus was on systems with index 0 with an emphasis on exponential stability and dichotomy.

We also add the following remark concerning the result in thm:IV-closed.
By cor:M_0-injective we know that M_0(M_0,M_1)→ H
is injective. If (M_0,M_1) is closed, it follows that the
operator C(C)⊆(M_0,M_1)→(M_0,M_1)
given by 

    (C)    U_0∈(M_0,M_1)M_1U_0∈ M_0[(M_0,M_1)],
    
    CU_0    M_0^-1M_1U_0  (U_0∈(C))

is well-defined and closed. Using this operator, C, thm:IV-closed
states that if (M_0,M_1) is closed then -C generates a
C_0-semigroup on (M_0,M_1). The precise statement can
be found in <cit.>. Moreover, C is
bounded if _(M_0,M_1) is closed (cf. exer:IV_k_closed).



§ EXERCISES

tocsectionExercises


Let M_0,M_1∈^n× n
such that (M_0,M_1) is regular and define the Wong sequence
(_j)_j∈_0 associated with (M_0,M_1). Moreover,
let P,Q∈^n× n, C∈^k× k, and N∈^(n-k)×(n-k)
be as in the quasi-Weierstra normal form for (M_0,M_1) with
N nilpotent (cf. prop:Weierstrass_normal_form). We
decompose a vector x∈^n into x̌∈^k and x̂∈^n-k
such that x=(x̌,x̂). Prove that 

    x∈_j⇔Q̂^̂-̂1̂x̂∈ N^j  (j∈_0).

Moreover, show that for each z∈ρ(M_0,M_1) we have

    _j=((zM_0+M_1)^-1M_0)^j  (j∈_0).




 Let E∈^n× n. We set k(E,1),
where 1 denotes the identity matrix in ^n× n. A matrix
X∈^n× n is called a Drazin inverse of E if
the following properties hold:


  * EX=XE,

  * XEX=X,

  * XE^k+1=E^k.

Prove that each matrix E∈^n× n has a unique Drazin inverse.

Hint: For the existence consider the quasi-Weierstra form for (E,1).



 Let M_0,M_1∈^n× n with
(M_0,M_1) regular and M_0M_1=M_1M_0. Denote by M_0
the Drazin inverse of M_0 (see exer:Drazin-inverse).
Prove:


  * M_0M_1=M_1M_0,

  * M_0M_0=(M_0,M_1),

  * For all U_0∈(M_0,M_1) the solution U of eq:DAE
is given by 

    U(t)=^-tM_0M_1U_0  (t≥0).





 Let M_0,M_1∈^n× n
with (M_0,M_1) regular. Prove that there exist two matrices
E,A∈^n× n with (E,A) regular and EA=AE such that


  * (E,A)=(M_0,M_1),

  * U solves the initial value problem eq:DAE for the
matrices M_0,M_1 if and only if U solves the initial value
problem eq:DAE for the matrices E,A with the same
initial value U_0∈(M_0,M_1).




 We consider the following electrical
circuit (see Figure <ref>) with a resistor with resistance
R>0, an inductor with inductance L>0 and a capacitor with capacitance
C>0. We denote the respective voltage drops by v_R,v_L and v_C.
Moreover, the current is denoted by i. The constitutive relations
for resistor, inductor and capacitor are given by 

    Ri     =v_R,
    
    Li'     =v_L,
    
    Cv_C'     =i,

respectively. Moreover, by Kirchhoff's second law we have 

    v_R+v_C+v_L=0.

Write these equations as a differential algebraic equation and compute
the index and the space of consistent initial values. Moreover, compute
the solution for each consistent initial value for R=2 and C=L=1.



 Prove the assertions <ref> to <ref> in
lem:properties-Wong.



 Let M_0,M_1∈ L(H).


  * Assume that ρ(M_0,M_1)∅. Prove that for
each k∈ the space _k is closed if and only
if M_0[_k-1] is closed.

  * Assume that (M_0,M_1) is regular with (M_0,M_1)≥1.
Prove that if _(M_0,M_1) is closed then the operator

    M_0|__(M_0,M_1)_(M_0,M_1)→ M_0[_(M_0,M_1)-1]

is an isomorphism.




[heading=subbibliography]



CHAPTER: EXPONENTIAL STABILITY OF EVOLUTIONARY EQUATIONS


In this chapter we study the exponential stability of evolutionary
equations. Roughly speaking, exponential stability of a well-posed
evolutionary equation 

    (νM(ν)+A)U=F

means that exponentially decaying right-hand sides F lead to exponentially
decaying solutions U. The main problem in defining the notion of
exponential decay for a solution of an evolutionary equation is the
lack of continuity with respect to time, so a pointwise definition would not make sense
in this framework. Instead, we will use our exponentially weighted
spaces (;H), but this time for negative ν, and define
the exponential stability by the invariance of these spaces under
the solution operator associated with the evolutionary equation under
consideration. 



§ THE NOTION OF EXPONENTIAL STABILITY


Throughout this section, let H be a Hilbert space, M(M)⊆→(H) a
material law and A(A)⊆ H→ H
a skew-selfadjoint operator. Moreover, we assume that there exist
ν_0>M and c>0 such that 

    zM(z)≥ c  (z∈_≥ν_0).

By Picard's theorem (thm:Solution_theory_EE) we know that for ν≥ν_0 the operator

    S_ν(νM(ν)+A)^-1∈((;H))

is causal and independent of the particular choice of ν. We now define the notion of exponential stability.

We call the solution operators (S_ν)_ν≥ν_0



exponentially stableexponentially stable with decay
rate ρ_0>0 if for all ρ∈0ρ_0 and ν≥ν_0
we have 

    S_νF∈-ρ(;H)  (F∈(;H)∩-ρ(;H)).



We emphasise that the definition
of exponential stability does not mean that the evolutionary equation
is just solvable for some negative weights. Indeed, if we consider

H=, A=0 and M(z)=1 for z∈
we obtain that the corresponding evolutionary equation 

    νU=F

is well-posed for each ν0. However, we also place a demand for
causality on our solution operator. Thus, we only have to consider parameters ν>0.
We obtain the solution U by

    U(t)=∫_-∞^tF(s)ṣ.

As it turns out, the problem eq:dtnuUeqF is not exponentially stable. Indeed, for F_[0,1]∈⋂_ν∈() the solution U is given by

    U(t)=
    0     if t<0,
    
    t     if 0≤ t≤1,
    
    1     if t>1,

which does not belong to the space -ρ() for any ρ>0. 


We first show that the aforementioned notion of exponential stability
also yields a pointwise exponential decay of solutions if we assume
more regularity for our source term F. 

 Let (S_ν)_ν≥ν_0 be exponentially stable with decay
rate ρ_0>0, ν≥ν_0, ρ∈0ρ_0 and F∈(ν)∩(-ρ).
Then U S_νF
is continuous and satisfies 

    U(t)^ρ t→0  (t→∞).




We first note that νF=-ρF by exer:derivative_invariant.
Moreover, since S_ν is a material law operator (i.e., S_ν = S(ν) for some material law S; see rem:somlo) we have

    S_νν⊆νS_ν.

Thus, in particular, we have

    S_ννF=νS_νF=νU;

that is, U∈(ν). Moreover, since νF=-ρF∈-ρ(;H),
we infer also U,νU∈-ρ(;H) by exponential
stability. By exer:derivative_invariant this yields 
U∈(-ρ) with -ρU=νU. The assertion
now follows from the Sobolev embedding theorem (thm:Sobolev_emb
and cor:vanish_at_inf).




§ A CRITERION FOR EXPONENTIAL STABILITY OF PARABOLIC-TYPE EQUATIONS


In this section we will prove a useful criterion for exponential stability
of a certain class of evolutionary equations. The easiest example
we have in mind is the heat equation with homogeneous Dirichlet
boundary conditions, which can be written as an evolutionary equation
of the form (cf. thm:wp_heat)

    (ν[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0   ; _0  0 ])[ θ; q ]=[ Q; 0 ]

in (;H), where H=Ł(Ω)⊕Ł(Ω)^d with Ω⊆^d
open, and a∈ L(Ł(Ω)^d) with 

    a≥ c

for some c>0 which models the heat conductivity, and ν>0.





Let H_0,H_1 be Hilbert
spaces and C(C)⊆ H_0→ H_1 a densely defined closed
linear operator which is boundedly invertible. Moreover, let M_0∈(H_0)
be selfadjoint with 

    M_0≥ c_0

for some c_0>0 and M_1(M_1)⊆→(H_1) be
a material law satisfying M_1< -ρ_1 for some ρ_1>0 and 

    ∃  c_1>0 ∀ z∈_>-ρ_1:  M_1(z)≥ c_1.

Then

    S_ν(ν[ M_0   0;   0   0 ]+[      0      0;      0 M_1(ν) ]+[    0 -C^∗;    C    0 ])^-1∈((;H_0⊕ H_1))

for each ν>0. Moreover, for all ν_0>0 the family
(S_ν)_ν≥ν_0 is exponentially stable with decay rate ρ_0min{ρ_1,c_1/(M_1^2_∞,_>-ρ_1M_0C^-1^2)}.



In order to prove this theorem, we need some prerequisites. We begin
with the well-known theorem of Riemann on removable singularities
for meromorphic functions with values in a Banach space.

 Let X be a Banach space, Ω⊆
open, z_0∈Ω and f:Ω∖{z_0}→ X analytic
and bounded. Then z_0 is a removable sngularity of f, i.e.,
there is an analytic function F:Ω→ X with F(z)=f(z) for
each z∈Ω∖{z_0}.



We consider the function 

    h(z)
    (z-z_0)^2f(z)     if z z_0,
    
    0     if z=z_0,
      (z∈Ω).

Then h is continuously differentiable in z_0 with h'(z_0)=0
since 

    h(z)/z-z_0=(z-z_0)f(z)→0  (z→ z_0)

by the boundedness of f. Hence, h is holomorphic and thus, we
can expand h in a power series in a neighbourhood of z_0,
i.e. 

    h(z)=∑_n=0^∞a_n(z-z_0)^n  (z∈z_0r)

for some r>0 small enough and a_n∈ X for n∈. Since
h(z_0)=h'(z_0)=0, we infer that a_0=a_1=0. Thus, we
have 

    f(z)=h(z)/(z-z_0)^2=∑_n=0^∞a_n+2(z-z_0)^n  (z∈z_0r∖{z_0}),

where the right hand side is also analytic in z_0. 



In order to prove this theorem we need a preparatory result.



Assume the hypotheses of thm:exp_stability_parabolic.
Then for each z∈_>-ρ_0 the operator 

    T(z)[   zM_0      0;      0 M_1(z) ]+[    0 -C^∗;    C    0 ](C)×(C^∗)⊆ H_0⊕ H_1→ H_0⊕ H_1

is boundedly invertible. Moreover, 

    sup_z∈_≥-ρT(z)^-1<∞

for each ρ<ρ_0.



Let z∈_≥-ρ for some ρ<ρ_0. We note that
M_1(z) is boundedly invertible with M_1(z)^-1≤ 1/c_1 (see prop:block_op_realinv<ref>) and (C^∗)^-1=(C^-1)^∗∈(H_0,H_1) (see lem:adj and lem:bdds). The beginning of the proof deals with a reformulation of T(z). For this, let u, f∈ H_0, v,g∈ H_1. Then, by definition, (u,v)∈(T(z))=(C)×(C^*) and T(z)(u,v)=(f,g) if and only if v∈(C^*) and u∈(C) together with

    zM_0 u - C^*v    = f
    
     Cu + M_1 (z)v    =g.

Since both C^* and M_1(z) are continuously invertible, we obtain equivalently u∈(C) together with

    z(C^*)^-1M_0 u - v    = (C^*)^-1f 
    
       M_1 (z)^-1Cu + v    =M_1 (z)^-1g.

Adding the latter two equations and retaining the first equation, we obtain the following equivalent system subject to the condition u∈(C)

    v    =z(C^*)^-1(M_0 u - f)∈(C^*), 
    
        (z(C^*)^-1M_0C^-1  +   M_1 (z)^-1)Cu     =M_1 (z)^-1g + (C^*)^-1f.





























We now inspect the operator S(z) z(C^-1)^∗M_0C^-1+M_1(z)^-1∈(H_1).
By prop:block_op_realinv for x∈ H_1 we estimate 

    xS(z)x    =C^-1xzM_0C^-1x+xM_1(z)^-1x
       ≥-ρM_0C^-1^2x^2+c_1/M_1(z)^2x^2
       ≥(c_1/M_1_∞,_>-ρ_1^2-ρM_0C^-1^2)_μx^2.

Since ρ<ρ_0 and by the definition of ρ_0
we infer that μ>0. Hence, S(z) is boundedly invertible with

    S(z)^-1≤1/μ.

We now set 

    u     C^-1S(z)^-1((C^∗)^-1f+M_1(z)^-1g)∈(C),
    
    v     z(C^*)^-1(M_0 u - f)∈(C^*).

By the first part of the proof we have that (u,v)
is the unique solution of T(z)(u,v)=(f,g). Moreover, we can estimate

    u   ≤C^-11/μ((C^∗)^-1f+1/c_1g), and 
    v   ≤1/c_1(g+Cu) ≤1/c_1(g+1/μ((C^∗)^-1f+1/c_1g)),

which proves that T(z) is boundedly invertible with 

    sup_z∈_≥-ρT(z)^-1<∞.*





Let H H_0⊕ H_1.
We set 

    M(z)[        M_0          0;          0 z^-1M_1(z) ]  (z∈(M_1)∖{0}).

Let ν>0. Then

    ∀ z∈_≥ν:  zM(z)≥min{ν c_0,c_1}

and hence, the first assertion of the theorem follows from thm:Solution_theory_EE.


Next, we focus on exponential stability. For this note that (S_ν)_ν≥ν_0 is exponentially stable if S_ν∈((;H)∩-ρ(;H)) for all ρ∈0ρ_0 and ν≥ν_0. For this, by lem:simple_fcts_compact_spt,  it suffices to show the corresponding estimate on S_(;X). Therefore, we let F∈ S_(;H)⊆(;H) and define

    U(ν[ M_0   0;   0   0 ]+[      0      0;      0 M_1(ν) ]+[    0 -C^∗;    C    0 ])^-1F.
 Since the estimate U_≤C̃F_ for some C̃≥ 0 is a consequence of the continuity of the solution operator, it is left to show that for all ρ∈0ρ_0 there exists C≥ 0 such that U_-ρ≤ CF_-ρ. 

For this, we observe that since F has compact support, we find  n∈ such that F⊆_≥-n. Then U∈(_≥-n;H) by causality. Since 
    (ν[ M_0   0;   0   0 ]+[      0      0;      0 M_1(ν) ]+[    0 -C^∗;    C    0 ])^-1
 is a material law operator (induced by the material law T(·)^-1 from lem:pointwise_inverse), it commutes with the material law operator τ_-n (see rem:mlohom for the commutation and rem:somlo as well as exa:material_law_revisited for the material law operator properties). Thus, we obtain

    τ_-nU=(ν[ M_0   0;   0   0 ]+[      0      0;      0 M_1(ν) ]+[    0 -C^∗;    C    0 ])^-1τ_-nF

and  τ_-nF∈-ρ(_≥0;H) by assumption.
Hence, cor:Lapalce_unitary yields ℒτ_-nF∈(_>-ρ;H). Let T(·) be as in lem:pointwise_inverse. We note here that thus 
    T(·)^-1(ν)=(ν[ M_0   0;   0   0 ]+[      0      0;      0 M_1(ν) ]+[    0 -C^∗;    C    0 ])^-1.

Since T(·)^-1 has a bounded analytic extension to _>-ρ, we obtain T(·)^-1ℒτ_-nF ∈(_>-ρ;H).
Since τ_-nU∈(_≥0;H) and 

    (ℒτ_-nU )(z)=T(z)^-1(ℒτ_-nF )(z)  (z∈_>ν),

we conclude that τ_-nU ∈-ρ(_≥0;H) by cor:Lapalce_unitary. The boundedness of T(·)^-1 on _>-ρ implies the desired continuity estimate. In order to see this, let Csup_z∈_≥-ρT(z)^-1. Using that ℒ_-ρ is unitary from -ρ(;H) to Ł(;H),
we compute 

    U_-ρ =ℒ_-ρU_Ł 
     =T(ı-ρ)^-1ℒ_-ρF_Ł≤ Cℒ_-ρF_Ł
     =CF_-ρ,

which is the desired continuity estimate.




§ THREE EXPONENTIALLY STABLE MODELS FOR HEAT CONDUCTION





 §.§ The Classical Heat Equation


We recall the classical heat equation (cf. thm:wp_heat)
on an open subset Ω⊆^d consisting of two equations,
the heat-flux balance 

    ∂_t θ+ q=f

and Fourier's law 

    q=-aθ,

where f is a given source term and a∈(Ł(Ω)^d) is
an operator modelling the heat conductivity of the underlying medium.
We will impose Dirichlet boundary conditions which will be incorporated
in our equation by replacing the operator  by _0
in Fourier's law (cf. sec:First-order-Sobolev).

In order to apply thm:exp_stability_parabolic we need
that _0 is boundedly invertible in some sense. This can
be shown using Poincaré's inequalityPoincaré's inequality.

 Let Ω⊆^d be open and contained
in a slab; that is, there exist 
e∈^d with e=1 and a,b∈, a<b such that

    Ω⊆x∈^da<ex<b.

Then for each u∈(_0) we have

    u_Ł(Ω)≤(b-a)_0u_Ł(Ω)^d.




Without loss of generality, let e=(1,0,…,0).
Recall that, by definition, (Ω) is a core for
_0. Thus, it suffices to prove the assertion for functions
in (Ω). Let φ∈(Ω).
We identify φ with its extension by 0 to the whole of
^d. By the fundamental theorem of calculus, we may compute

    φ(x)=∫_a^x_1∂_1φ(s,x_2,…,x_d)ṣ   (x∈Ω).

Hence, by the Cauchy–Schwarz inequality and Tonelli's theorem

    ∫_Ωφ(x)^2x̣    =∫_Ω∫_a^x_1∂_1φ(s,x_2,…,x_d)ṣ^2x̣
       ≤∫_Ω(b-a)∫_a^b(∂_1φ(s,x_2,…,x_d))^2ṣx̣
     = (b-a)^2∫_Ω∂_1φ(x)^2x̣
       ≤(b-a)^2_0φ_Ł(Ω)^d^2,

which shows the assertion. 


 Under the assumptions of prop:Poincare
the operator _0 is one-to-one and (_0) is
closed.



The injectivity follows immediately from Poincaré's inequality. To
prove the closedness of (_0), let (u_k)_k∈
in (_0) with _0u_k→ v in Ł(Ω)^d
for some v∈Ł(Ω)^d. By Poincaré's inequality, we infer
that (u_k)_k∈ is a Cauchy-sequence in Ł(Ω) and
hence convergent to some u∈Ł(Ω). By the closedness of _0
we obtain u∈(_0) and v=_0u∈(_0). 


We need another auxiliary result which is interesting in its own
right.


 Let H be a Hilbert space and
V⊆ H a closed subspace. We denote by 

    ι_V V→ H,   x↦ x

the canonical embedding of V into H. Then ι_Vι_V^∗ H→ H
is the orthogonal projection on V and ι_V^∗ι_V V→ V
is the identity on V. 



The proof is left as exer:embeddings.


We now come to the exponential stability of the heat equation. First,
we need to formulate both the heat-flux balance and Fourier's law
as a suitable evolutionary equation. For doing so, we assume that
Ω⊆^d is open and contained in a slab. Then (_0)
is closed by cor:range(grad) closed. It is clear that
we can write Fourier's law as 

    q=-a_0θ=-aι_(_0)ι_(_0)^∗_0θ.

Hence, defining q̃ι_(_0)^∗q
and ãι_(_0)^∗aι_(_0)∈((_0)),
we arrive at 

    q̃=-ãι_(_0)^∗_0θ.

Moreover, since (_0)^=(), we derive from
the heat-flux balance 

    f     =∂_tθ+ q
     =∂_tθ+ι_(_0)q̃

and hence, assuming that ã is invertible, we may write both
equations with the unknowns (θ,q̃) as an evolutionary
equation in (;H) for ν>0, where HŁ(Ω)⊕(_0). This yields 

    (ν[ 1 0; 0 0 ]+[    0    0;    0 ã^-1 ]+[          0     ι_(_0); ι_(_0)^∗_0          0 ]) [  θ; q̃ ]=[ f; 0 ].


For notational convenience, we set

    Cι_(_0)^∗_0(_0)⊆Ł(Ω)→(_0).



 Let Ω⊆^d be open and contained
in a slab and C as above. Then C is densely defined, closed
and boundedly invertible. Moreover 

    C^∗=-ι_(_0).




The proof is left as exer:closed_range.


Let Ω⊆^d be open and contained
in a slab, a∈(Ł(Ω)^d), and c_1>0 such that

    a≥ c_1.

Then ãι_(_0)^∗aι_(_0)
is boundedly invertible and the solution operators associated with eq:heat_modified are exponentially
stable.



For x∈(_0) we have 

    xãx_(_0)    =ι_(_0)xaι_(_0)x_Ł(Ω)^d
       ≥ c_1ι_(_0)x_Ł(Ω)^d^2
     =c_1x_(_0)^2,

and thus, ã is boundedly invertible. Hence, eq:heat_modified
is an evolutionary equation of the form considered in thm:exp_stability_parabolic
with M_0, M_1(z)ã^-1 for z∈
and C given by eq:C. Since ã^-1≥c_1/ã^2,
thm:exp_stability_parabolic is applicable and we derive
the exponential stability.




 §.§ The Heat Equation with Additional Delay


Again we consider the heat equation, but now we replace Fourier's
law by

    q=-a_1_0θ-a_2τ_-h_0θ

for some operators a_1,a_2∈(Ł(Ω)^d) and h>0.
As above, we assume that Ω⊆^d is open and contained in a slab. We may
introduce q̃ι_(_0)^∗q and
ã_jι_(_0)^∗a_jι_(_0)∈(Ł(Ω)^d)
for j∈{1,2}. Moreover, we assume that there exists c>0 such that

    a_1≥ c.

By lem:delayInv there exists ν_0>0 such that the operator ã_1+ã_2τ_-h
is boundedly invertible in (;(_0)) and its inverse is uniformly strictly positive definite for each ν≥ν_0. Hence, we may write the heat equation with additional
delay as an evolutionary equation of the form 

    (ν[ 1 0; 0 0 ]+[                0                0;                0 (ã_1+ã_2τ_-h)^-1 ]+[    0 -C^∗;    C    0 ])[  θ; q̃ ]=[ f; 0 ]

with C given by eq:C. 

Let Ω⊆^d be open and contained in a slab, h>0, a_1,a_2∈(Ł(Ω)^d), and c>0 such that

    a_1≥ c

and  a_2<c. Then the solution operators (S_ν)_ν≥ν_0 associated with eq:heat_delay_exp_stability
are exponentially stable.



Note that ã_2≤a_2<c. We choose

    0<ρ_1<1/hlogc/ã_2.

Then we estimate for z∈_>-ρ_1

    x(ã_1+ã_2^-zh)x_(_0)≥(c-ã_2^ρ_1h)x_(_0)^2.

By the choice of ρ_1, we infer  c̃(c-ã_2^ρ_1h)>0.
Hence, 

    M_1(z)(ã_1+ã_2^-hz)^-1  (z∈_>-ρ_1)

is well-defined and satisfies 

    M_1(z)≥ c_1  (z∈_>-ρ_1)

for some c_1>0 by prop:block_op_realinv. Thus, thm:exp_stability_parabolic
is applicable and yields the exponential stability of eq:heat_delay_exp_stability. 




 §.§ A Dual Phase Lag Model


In this last variant of heat conduction, we replace Fourier's law
by 

    (1+s_q∂_t)q=(1+s_θ∂_t)_0θ,

where s_q,s_θ>0 are the so-called “phases” (cf. sec:Dual-phase-lag,
where a different type of dual phase lag model is studied). The latter
equation can be reformulated as 

    (1+s_qν)(1+s_θν)^-1q=_0θ

for ν>0. Assuming that Ω⊆^d is open and contained in a slab, and defining q̃ι_(_0)^∗q,
the dual phase lag model may be written as 

    (ν[ 1 0; 0 0 ]+[                   0                   0;                   0 (1+s_qν)(1+s_θν)^-1 ]+[    0 -C^∗;    C    0 ])[  θ; q̃ ]=[ f; 0 ]

with C given by eq:C.

Let Ω⊆^d be open and contained in a slab, ν_0>0. Moreover,
let s_θ>s_q>0. Then the solution operators (S_ν)_ν≥ν_0 associated with eq:dual_phase_exp are
exponentially stable.



Again, we note that eq:dual_phase_exp is of the form
considered in thm:exp_stability_parabolic with M_0
and 

    M_1(z)1+s_qz/1+s_θz  (z∈∖{-s_θ^-1}).

Setting μs_q/s_θ<1 we compute 

    M_1(z)=(μ+(1-μ)/1+s_θz)=μ+(1-μ)1+s_θ z/|1+s_θz|^2≥μ  (z∈_>-s_θ^-1).

Thus, thm:exp_stability_parabolic is applicable and hence,
the claim follows.




§ COMMENTS


The results of this chapter are based on the results obtained in <cit.>.
There, Laplace transform techniques are used to characterise the exponential
stability of evolutionary equations in a slightly more general setting.
Moreover, besides parabolic-type equations, also hyperbolic-type equations
are considered, such as the damped wave equation or the equations of
visco-elasticity.

The exponential stability of partial differential equations is a well-studied
field. In particular, in the framework of C_0-semigroups, where
the exponential stability is defined via pointwise estimates due
to the continuity of solutions, several results are known. We just
mention Datko's theorem <cit.> (see also <cit.>),
which states that a C_0-semigroup is exponentially stable if
and only if the solution operator associated with the equation 

    (ν+A)U=F

leaves L_p(_≥0;H) invariant for some (or equivalently
all) p∈1∞. As it turns out, the latter is equivalent
to the invariance of -ρ(;H) for some ρ>0 and thus,
our notion of exponential stability coincides with the usual one used
in the theory of C_0-semigroups. Another important theorem on the exponential
stability of C_0-semigroups on Hilbert spaces is the Theorem
of Gearhart–Prüß <cit.> (see also <cit.>),
where the exponential stability of a C_0-semigroup is characterised
in terms of the resolvent of its generator.

Besides the exponential stability, which is the only type of stability
studied so far within the current framework, different kinds of
asymptotic behaviours were considered for C_0-semigroups. We
just mention the celebrated Arendt–Batty–Lyubich–Vu theorem <cit.>
on strong stability of C_0-semigroups or the Theorem of Borichev–Tomilov
<cit.> on the polynomial stability of C_0-semigroups
on Hilbert spaces.



§ EXERCISES

tocsectionExercises


 Let H be a Hilbert space, ν,ρ∈
and u∈ L_1,(;H). Prove the following statements:


  * If u∈(ν)∩(ρ) then νu=ρu. 

  * If u∈(ν) such that u,νu∈ρ(;H)
then u∈(ρ).





Prove lem:embeddings_projections.



Let H_0,H_1 be Hilbert spaces and
A(A)⊆ H_0→ H_1 a densely defined closed
linear operator. Moreover, we assume that A has closed
range. Show that the adjoint of the operator ι_(A)^∗A(A)⊆ H_0→(A)
is given by A^∗ι_(A). If additionally A is one-to-one, show that ι_(A)^∗ A is boundedly invertible.



 Let Ω⊆^d be open and contained
in a slab. We consider the heat conduction with a memory term given
by the equations 

    νθ+ q     =f,
    
    q     =-(-k∗)_0θ,

where k∈ L_1,-ρ_1(_≥0;) for some ρ_1>0
with

    ∫_0^∞k(t)ṭ<1.

Write eq:heat_integro as a suitable evolutionary equation
and prove that this equation is exponentially stable. 



 Let A∈^n× n for some
n∈ and consider the evolutionary equation 

    (ν+A)U=F.

Prove that the solution operators associated with this problem are exponentially stable if and only if A has only eigenvalues with strictly positive real part. 



 Let Ω⊆^d be open. 



  * Let φ∈(Ω)^d. Prove Korn's
inequality Korn's inequality

    φ_Ł(Ω)_^d× d^2≥1/2∑_j=1^dφ_j_Ł(Ω)^d^2.


  * Use Korn's inequality to prove that for u∈Ł(Ω)^d
we have 

    u∈(_0)  ⟺  ∀ j∈{1,…,d}: u_j∈(_0).

Moreover, show that in either case 

    1/2∑_j=1^d_0u_j_Ł(Ω)^d^2≤_0u_Ł(Ω)_^d× d^2≤∑_j=1^d_0u_j_Ł(Ω)^d^2.


  * Let now Ω be contained in a slab. Prove
that _0 is one-to-one and has closed range. 




  Let Ω⊆^d open and a∈(Ł(Ω)^d) with a≥ c>0.
  
   
  * Let ν>0 and f∈(;Ł(Ω)). 
   Moreover, assume that Ω is contained in a slab and define ãι_(_0)^∗ a ι_(_0).
   Let θ∈(;Ł(Ω)), q∈(;Ł(Ω)^d) satisfy
   
    (ν[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0   ; _0  0 ]) [ θ; q ]=[ f; 0 ]

   and θ̃∈(;Ł(Ω)), q̃∈(;(_0)) satisfy
   
    (ν[ 1 0; 0 0 ]+[    0    0;    0 ã^-1 ]+[          0     ι_(_0); ι_(_0)^∗_0          0 ]) [ θ̃; q̃ ]=[ f; 0 ].

Show that (θ,ι_(_0)^∗ q)=(θ̃,q̃ ).

  * Let Ω be bounded and consider the evolutionary equation

    (ν[ 1 0; 0 0 ]+[    0    0;    0 a^-1 ]+[  0 _0;     0 ]) [ θ; q ]=[ f; 0 ].

Show that the associated solution operators are not exponentially stable.
  





[heading=subbibliography]



CHAPTER: BOUNDARY VALUE PROBLEMS AND BOUNDARY VALUE SPACES


This chapter is devoted to the study of inhomogeneous boundary value
problems. For this, we shall reformulate the boundary value problem
again into a form which fits within the general framework of evolutionary equations.
In order to have an idea of the type of boundary values which make sense
to study, we start off with a section that deals with the boundary
values of functions in the domain of the gradient operator defined
on a half space in ^d (for d=1 we have Ł(^d-1)=).



§ THE BOUNDARY VALUES OF FUNCTIONS IN THE DOMAIN OF THE GRADIENT


In this section we let Ω^d-1×0 and f∈ H^1(Ω); our aim is
to make sense of the function ^d-1∋x̌↦ f(x̌,0).
Note that this makes no sense if we only assume f∈Ł(Ω)
since ^d-1×{0}=∂Ω is a set of
(d-dimensional) Lebesgue-measure zero. However, if we assume f
to be weakly differentiable, something more can be said and the boundary
values can be defined by means of a continuous extension of the so-called
trace map. In order to properly formulate this, we need the following density
result.

 The set 𝒟ϕΩ→∃ψ∈(^d)ψ|_Ω=ϕ
is dense in the space H^1(Ω).


We will need a density result for H^1(^d) first.


 (^d) is dense in H^1(^d).



Let f∈ H^1(^d). We first show that f can be approximated
by functions with compact support. For this let ϕ∈(^d)
with the properties 0≤ϕ≤1, ϕ=1 on 01/2
and ϕ=0 on ∖01. For all k∈
we put ϕ_kϕ(·/k) and f_kϕ_kf∈Ł(^d).
Then f_k has support contained in 0k. The dominated
convergence theorem implies that f_k→ f in Ł(^d)
as k→∞. Next, let ψ∈(^d)^d and compute
for all k∈

    -f_kψ    =-ϕ_kfψ
     =-fϕ_kψ
     =-f(ϕ_kψ)-(ϕ_k)·ψ
        =-f(ϕ_kψ)+fϕ_kψ
     =( f)ϕ_k+1/kf(ϕ)(·/k)ψ,

which shows that f_k∈()=H^1(^d) and f_k=( f)ϕ_k+1/kf(ϕ)(·/k).
From this expression of f_k we observe f_k→ f in Ł(^d)^d by dominated convergence. Hence, f_k→ f in ()=H^1(^d). 

To conclude the proof of this lemma it suffices to revisit exer:C_cinfty dense.
For this, let (ψ_k)_k in (^d) be a δ-sequence.
Then, by exer:C_cinfty dense, we infer ψ_k∗ f→ f in Ł(^d)
as k→∞ and hence, by exer:convSmooth, it follows also that
(ψ_k∗ f)=ψ_k∗ f→ f
(note the component-wise definition of the convolution). A combination
of the first part of this proof together with an estimate for the
support of the convolution (see again exer:C_cinfty dense)
yields the assertion.



Let f∈ H^1(Ω). The approximation of f by functions in 𝒟 is done in two steps.
First, we shift f in the negative e_d-direction to avoid the boundary, and then we convolve the shifted f to obtain smooth approximants in 𝒟.

Let f̃∈Ł(^d) be the extension of f by zero.
Put e_d(δ_jd)_j∈{1,… d}, the d-th
unit vector. Then for all τ>0 we have Ω+τ e_d⊆Ω
and, thus by exer:H1translate, we deduce  f_τf̃(·+τ e_d)|_Ω→ f in H^1(Ω) as τ→ 0.
Thus, it suffices to approximate f_τ for τ>0.

Let τ>0 and let (ψ_k)_k in (^d) be a δ-sequence.
Then ψ_k∗f̃(·+τ e_d)∈ H^1(^d),
by exer:convSmooth. Define f_k,τ(ψ_k∗f̃(·+τ e_d))|_Ω.
Then we obtain that f_k,τ→ f_τ in
H^1(Ω) as k→∞. Indeed, the only thing left to
prove is that f_k,τ→ f_τ in Ł(Ω)^d
as k→∞. For this, we denote by g the extension of f
by 0. Since g∈Ł(^d)^d it suffices to show that f_k,τ=ψ_k∗ g_τ
on Ω for all large enough k∈, where g_τ=g(·+τ e_d).
Let k>1/τ. Then for all x∈Ω
and y∈ψ_k⊆-1/k1/k^d we infer x-y+τ e_d∈Ω.
In particular, f(·-y+τ e_d)∈ H^1(Ω) and f(·-y+τ e_d)=g(·-y+τ e_d).
Take η∈(Ω)^d and compute

    -f_k,τη_Ł(Ω)    =-∫_Ω∫_^dψ_k(x-y)f̃(y+τ e_d)^*ỵη(x)x̣
        =-∫_Ω∫_^dψ_k(y)f̃(x-y+τ e_d)^*ỵη(x)x̣
        =-∫_Ω∫_-1/k1/k^dψ_k(y)f(x-y+τ e_d)^*ỵη(x)x̣
        =-∫_-1/k1/k^dψ_k(y)f(·-y+τ e_d)η_Ł(Ω)ỵ
        =∫_-1/k1/k^dψ_k(y)g(·-y+τ e_d)η_Ł(Ω)^dỵ
        =ψ_k∗ g_τη_Ł(Ω)^d.

As ψ_k∗f̃(·+τ e_d)∈ H^1(^d), we
conclude the proof using lem:CinfwholespaceDense.

With these preparations at hand, we can define the boundary trace
of H^1(Ω). 

 The operator

    γ𝒟⊆ H^1(Ω)    →Ł(^d-1)
    
    f    ↦(^d-1∋x̌↦ f(x̌,0))

is continuous, densely defined and, thus, admits a unique continuous
extension to H^1(Ω) again denoted by γ. Moreover, we have

    γ f_Ł(^d-1)≤(2f_Ł(Ω) f_Ł(Ω)^d)^1/2≤f_H^1(Ω)  (f∈ H^1(Ω)).

 



Note that γ is densely defined by thm:CinfbdryDense. Let f∈(^d) and
x̌∈^d-1. Let R>0 be such that f⊆0R.
Then

    ∫_^d-1f(x̌,0)^2̣̌x    = -∫_^d-1∫_0^R∂_df(x̌,x̂)^2̣̂x̣̌x
        = -∫_Ω(f(x)^*∂_df(x)+∂_df^*(x)f(x))x̣
       ≤2f_Ł(Ω) f_Ł(Ω)^d.

The remaining inequality follows from 2ab≤ a^2+b^2 for all a,b∈.


Except for one spatial dimension, where the boundary trace can be obtained
by point evaluation, the boundary trace γ does not map onto
the whole of Ł(^d-1). Hence, in order to define the space of
all possible boundary values for a function in H^1 one uses a
quotient construction: we set

    H^1/2(^d-1)γ ff∈ H^1(Ω)

 and endow H^1/2(^d-1) with the norm 

    γ f_H^1/2(^d-1)infg_H^1(Ω)g∈ H^1(Ω),γ g=γ f.

It is not difficult to see that H^1/2(^d-1) is unitarily
equivalent to (γ)^, where the orthogonal
complement is computed with respect to the scalar product in H^1(Ω).
Thus, H^1/2(^d-1) is a Hilbert space.


 The norm defined on the space H^1/2(^d-1) given above is not the standard norm defined on this space. Indeed, following <cit.> the usual norm is given by 
 
    (u_Ł(^d-1)^2+∫_^d-1∫_^d-1|u(x)-u(y)|^2/|x-y|^dx̣ỵ)^1/2

for u∈ H^1/2(^d-1). However, this norm turns out to be equivalent to the norm given above, see e.g. <cit.>. 

As the notation of this space suggests, it is can also defined as an interpolation space between H^1(^d-1) and Ł(^d-1), see <cit.>. 




§ THE BOUNDARY VALUES OF FUNCTIONS IN THE DOMAIN OF THE DIVERGENCE


Let Ω:=^d-1×0.
There is also a space of corresponding boundary traces for the divergence operator.
Similarly to the boundary values for the domain of the gradient operator, H^1(Ω), the construction of the boundary trace for H()-vector
fields rests on a density result. The proof can be done along the
lines of thm:CinfbdryDense and will be addressed in exer:divdesnity.

 𝒟^d is dense in H(,Ω). 


Equipped with this result, we can describe all possible boundary values
of H(,Ω). It will turn out that vector
fields in H(,Ω) have a well-defined normal
trace, which for Ω=^d-1×_>0 is just the negative of the last coordinate of the vector field.

 The operator 

    γ_n𝒟^d⊆ H(,Ω)    →(H^1/2(^d-1))' H^-1/2(^d-1)
    
    q    ↦(^d-1∋x̌↦ -q_d(x̌,0)),

is densely defined, continuous with norm bounded by 1 and has dense
range. Thus γ_n admits a unique extension to H(,Ω) again denoted by γ_n. 
Here, -q_d is the negative of the d-th component of q pointing into
the outward normal direction of Ω and -q_d is identified with the
linear functional 

    H^1/2(^d)∋γ f↦-q_d(·,0)γ f_Ł(^d-1).

Moreover, for all f∈() and q∈() we have

    qf+q f=(γ_n q)(γ f),

where we denoted the extension of γ_n again by γ_n.



Let f∈𝒟
and q∈𝒟^d. Then integration by parts yields

    qf+q f    =∫_Ω(q^∗ f)
     =∫_^d-1q^∗(x̌,0)f(x̌,0)-e_ḍ̌x
        =-∫_^d-1γ q_d^∗γ f
     =γ_nqγ f_Ł(^d-1)=(γ_n q)(γ f).

Hence, 

    γ_nqγ f_Ł(^d-1)≤q_H()f_H^1.

Since 𝒟 is dense in H^1(Ω), the inequality remains
true for all f∈ H^1(Ω). Thus,

    γ_nqγ f_Ł(^d-1)≤q_H()f_H^1  (f∈ H^1(Ω)).

Computing the infimum over all g∈ H^1(Ω) with γ g=γ f,
we deduce

    γ_nqγ f_Ł(^d-1)≤q_H()γ f_H^1/2(^d-1)  (f∈ H^1(Ω)).

Therefore γ_nq∈ H^-1/2(^d-1) and γ_nq_H^-1/2≤q_H(),
which shows continuity of γ_n. It is left to show
that γ_n has dense range. For this, take γ f∈ H^1/2(^d-1)
for some f∈ H^1(Ω) such that

    γ_ngγ f_Ł(^d-1)=0

for all g∈𝒟^d. Next, take g̃∈(^d-1)
and ψ∈() with ψ(0)=1. Then we set gΩ∋(x̌,x̂)↦ -e_dg̃(x̌)ψ(x̂) ∈𝒟^d
and note that γ_ng=g̃. Hence 

    γ fg̃_Ł(^d-1)=0  (g̃∈(^d-1)).

Thus, γ f=0, which implies that the range of γ_n
is dense, as H^1/2(^d-1) is a Hilbert space. The remaining
formula eq:ibptrace follows by continuously extending both the left- and right-hand
side of the integration by parts formula from the beginning of the
proof. Note that for this, we have used both thm:CinfbdryDense
and thm:divdensity.



Let f∈ H^1(Ω), q∈ H(,Ω). Then f∈(_0) if and only if γ f = 0, and 
q∈(_0) if and only if γ_nq=0.



We only show the statement for q. The proof for f is analogous.
If q∈(_0), then there exists a sequence (ψ_n)_n
in (Ω)^d such that ψ_n→ q in
H(,Ω) as n→∞. Thus, by continuity
of γ_n, we infer 0=γ_nψ_n→γ_nq.
Assume on the other hand that q∈() with γ_nq=0.
Using eq:ibptrace, we obtain for all f∈()

    qf+q f=0.

This equality implies that q∈(^*)=(_0), which shows the remaining assertion.

The remaining part of this section is devoted to showing that the continuous
extension of γ_n maps onto H^-1/2(^d-1).
For this we require the following observation, which will also be needed
later on.

Let U⊆^d be open. Then

    H_0(,U)^_H(,U)=q∈ H(,U) q∈ H^1(U),q= q.




Let q∈ H(,U). Then q∈ H_0(,U)^_H(,U)
if and only if for all r∈ H_0(,U) we have

    0     =rq_H(,U)
     =rq_Ł(U)^d+ r q_Ł(U)
     =rq_Ł(U)^d+_0r q_Ł(U).

The latter, in turn, is equivalent to q∈(_0^*)=()=H^1(U)
and - q=_0^* q=-q.


γ_n maps onto
H^-1/2(^d-1). In particular, we have

    q_H(,Ω)≤γ_nq_H^-1/2(^d-1)

for all q∈ H_0(,Ω)^_H(,Ω).



By thm:divetrace
it suffices to show that γ_n has closed
range. For this, it suffices to show that there exists c>0 such that

    q_H(,Ω)≤ cγ_nq_H^-1/2(^d-1)

for all q∈(γ_n)^_H(,Ω). 
By cor:CharDiv0, we obtain (γ_n)=H_0(,Ω).
Hence, by prop:BDdiv, we deduce that q∈(γ_n)^_H(,Ω)
if and only if q∈() and q= q. So,
assume that q∈() with q= q. Then
eq:ibptrace applied to q∈() and f= q∈() yields

    (γ_nq)(γ q)     = q q+ q q
     = q q+qq
        =q_H(,Ω)^2,

where we used q=q.
Hence 

    q_H(,Ω)^2   ≤γ q_H^1/2γ_nq_H^-1/2≤ q_H^1(Ω)γ_nq_H^-1/2
         =q_H(,Ω)γ_nq_H^-1/2

where we again used that q = q.
This yields the assertion.




§ INHOMOGENEOUS BOUNDARY VALUE PROBLEMS


Let Ω^d-1×0.
With the notion of traces we now have a tool at hand that allows us
to formulate inhomogeneous boundary value problems.  Here we focus on the
scalar wave type equation for given Neumann data g̃∈ H^-1/2(^d-1). We shall address
other boundary value problems in the exercises.
Let M(M)⊆→(Ł(Ω)×Ł(Ω)^d) be a material law with M<ν_0 for some ν_0∈. We assume that M satisfies the positive definiteness condition in thm:Solution_theory_EE; that is, we assume there exists c>0 such that for all z∈_≥ν_0 we have zM(z) ≥ c.
For ν≥ν_0 we want to solve

    (νM(ν)+[ 0  ;   0 ])[ v; q ]=[ 0; 0 ]    on Ω,
    γ_nq(t,·)=g̃    on ∂Ω for all t∈.

Let us reformulate this problem. Let ϕ∈ C^∞() such
that 0≤ϕ≤1 with ϕ=1 on 0∞ and ϕ=0
on -∞-1. We define the function g(t↦ϕ(t)g̃∈ H^-1/2(^d-1))∈⋂_ν>0(;H^-1/2(^d-1))
and consider

    (νM(ν)+[ 0  ;   0 ])[ v; q ]=[ 0; 0 ]    on Ω,
    γ_nq(t)=g(t)     for all t>0.

instead. 

 Let ν≥max{ν_0,0},ν≠0. Then eq:inhBVP admits a unique solution
(v,q)∈ H_ν^1(;([ 0  ;   0 ])). 



We start with the existence part. By thm:NeuTrOnto, we
find G̃∈ H(,Ω) such that γ_nG̃=g̃;
set Gϕ(·)G̃∈ H_ν^3(;H(,Ω)).
Consider the following evolutionary equation

    (νM(ν)+[  0 _0;     0 ])[ u; r ]
     =νM(ν)[  0; -G ]+[ - G;   0 ].

Note that the right-hand side is in H_ν^2(;Ł(Ω)×Ł(Ω)^d).
By thm:Solution_theory_EE, we obtain

    [ u; r ]    =(νM(ν)+[  0 _0;     0 ])^-1(νM(ν)[  0; -G ]+[ - G;   0 ])
       ∈ H_ν^1(;Ł(Ω)×Ł(Ω)^d)∩(;([ 0  ;   0 ])) .

Indeed, since the solution operator commutes with ν and the right-hand side lies in H_ν^2, it even follows that [ u; r ]∈ H_ν^2(;Ł(Ω)×Ł(Ω)^d).
 From the equality 

    (νM(ν)+[  0 _0;     0 ])[ u; r ]    =νM(ν)[  0; -G ]+[ - G;   0 ]

it follows that 

    [ u; r ]∈  H_ν^1(;([  0 _0;     0 ])).

Also, we deduce that 

    (νM(ν)+[ 0  ;   0 ])[   u; r+G ]=[ 0; 0 ].

Since r∈ H_ν^1(;(_0)), by cor:CharDiv0
and thm:Sobolev_emb we obtain

    γ_n((r+G)(t))=γ_nG(t)=g(t)    (t∈).

Hence, (u,r+G) solves eq:inhBVP. 

Next we address the uniqueness result. For this we note that a straightforward
computation shows

    [   v; q-G ]
      =(νM(ν)+[  0 _0;     0 ])^-1(νM(ν)[  0; -G ]+[ - G;   0 ]),

which coincides with the formula for (u,r+G). 

 The upshot of the rationale exemplified in the proof is that inhomogeneous
boundary value problems can be reduced to an evolutionary equation
of the standard form with non-vanishing right-hand side. Of course
the treatment of inhomogeneous Dirichlet data works along similar
lines. 



§ ABSTRACT BOUNDARY DATA SPACES


Of course inhomogeneous boundary value problems can be addressed for
other domains Ω than the half space ^d-1×0.
Classically, some more specific properties need to be imposed on the
description of the boundary ∂Ω. In this section, however,
we deviate from the classical perspective in as much as we like to
consider arbitrary open sets Ω⊆^d. For
this we introduce

    ()     =q∈ H(,Ω) q∈(), q=q,
    ()     =u∈ H^1(Ω) u∈(), u=u.

By prop:BDdiv and exer:H=00003DW+Kasuga,
these spaces are closed subspaces of H(,Ω) and H^1(Ω),
respectively, and therefore Hilbert spaces. Indeed, 

    ()=H_0(,Ω)^_H(,Ω)

and 

    ()=H^1_0(Ω)^_H^1(Ω).

Now, we are in a position to solve inhomogeneous boundary
value problems, where the trace mappings γ and γ_n
are replaced by the canonical orthogonal projections π_()
and π_() respectively; see exer:uniqunessBVP.
We devote the rest of this section to describe the relationship between
the classical trace spaces introduced before and the -spaces.
In the perspective outlined here, there is not much of a difference
between Neumann boundary values and Dirichlet boundary values. The
next result is an incarnation of this.

 We have 

    [()]⊆()  and  [()]⊆().

Moreover,
the mappings 

    _BD()   →(),
    
    u    ↦ u

 and 

    _BD()   →(),
    
    q    ↦ q

 are unitary, and _^*=_.



Let ϕ∈(). Then ϕ∈ H(,Ω)
and ϕ=ϕ. This implies ϕ∈()
and ϕ=ϕ, which yields ϕ∈().
Thus, _ is defined everywhere; interchanging the roles
of  and , we obtain _ is also defined
everywhere. We infer __=_()
and __=_() and thus _
is bijective with _^-1=_. It remains to show
that _ preserves the norm. For this we compute 

    _ϕ_ϕ_()    =ϕϕ_H()
        =ϕϕ_Ł(Ω)^d+ϕϕ_Ł(Ω)
        =ϕϕ_Ł(Ω)^d+ϕϕ_Ł(Ω)
        =ϕϕ_()=ϕϕ_(),

which implies that _ is unitary. Hence, _=_^-1=_^*.

It is also possible to show an `integration by parts' formula analogous
to eq:ibptrace for the abstract situation:

Let u∈ H^1(Ω) and q∈ H(,Ω).
Then

    qu_Ł(Ω)+q u_Ł(Ω)^d    =_π_()qπ_()u_()
        =π_()q_π_()u_().




We decompose u=u_0+u_1 and q=q_0+q_1 with u_0∈ H_0^1(Ω),
q_0∈ H_0(,Ω), u_1=π_()u
and q_1=π_()q. Then we obtain

    qu_Ł(Ω)+q u_Ł(Ω)^d
        =_0q_0u_Ł(Ω)+ q_1u_Ł(Ω)+q_0 u_Ł(Ω)^d+q_1 u_Ł(Ω)^d
        =q_0- u_Ł(Ω)^d+ q_1u_Ł(Ω)+q_0 u_Ł(Ω)^d+q_1 u_Ł(Ω)^d
        = q_1u_0_Ł(Ω)+ q_1u_1_Ł(Ω)+q_1 u_0_Ł(Ω)^d+q_1 u_1_Ł(Ω)^d
        =q_1-_0 u_0_Ł(Ω)^d+ q_1u_1_Ł(Ω)+q_1_0 u_0_Ł(Ω)^d+q_1 u_1_Ł(Ω)^d
        = q_1u_1_Ł(Ω)+q_1 u_1_Ł(Ω)^d
        = q_1u_1_Ł(Ω)+ q_1 u_1_Ł(Ω)^d= q_1u_1_().

The remaining equality follows from the unitarity of _ and
_^-1=_ by prop:divgradunit.

In view of prop:ibpabstract the proper replacement of γ_n
appears to be _π_() instead of just
π_(). Next, we show the equivalence of the
trace spaces for the half space and the abstract ones introduced in
this section.

Let Ω^d-1×0.
Then γ|_():()→ H^1/2(^d-1) and γ_n|_():()→ H^-1/2(^d-1)
are unitary mappings.



We begin with γ_n. We have shown in thm:divetrace
that γ_n|_() is continuous and
in thm:NeuTrOnto it has been shown that (γ_n|_())^-1
is continuous. Also the two norm inequalities have been established.

The injectivity of γ|_() follows from γ=H_0^1(Ω) by cor:CharDiv0.
All that remains simply relies upon recalling that H^1/2(^d-1) is isomorphic to (γ)^
with the orthogonal complement computed in H^1(Ω). 




§ ROBIN BOUNDARY CONDITIONS


The classical Robin boundary conditions involve both traces, the Dirichlet
trace γ and the Neumann trace γ_n. To motivate
things, let us again have a look at the case Ω=^d-1×0.
We consider the boundary condition for given q∈ H(,Ω)
and u∈ H^1(Ω)

    γ_nq+ıγ u=0,

in the sense that

    (γ_nq)(v)= -ıγ uv_Ł(^d-1)   (v∈ H^1/2(^d-1)).

Note that this is an implicit regularity statement as  γ_nq∈ H^-1/2(^d-1) is representable as an Ł(^d-1) function. The next result
asserts that an evolutionary equation with a spatial operator of the
type [ 0  ;   0 ] with the above Robin boundary condition fits into the setting rendered
by thm:Solution_theory_EE. In other words:

 Let Ω=^d-1×0. Then the
operator A(A)⊆Ł(Ω)^d+1→Ł(Ω)^d+1
with A⊆[ 0  ;   0 ] with domain

    (A)=(u,q)∈ H^1(Ω)× H(,Ω)γ_nq+ıγ u=0

is skew-selfadjoint.



Let (u,q),(v,r)∈ H^1(Ω)× H(,Ω). Then, by
eq:ibptrace we obtain

    [ 0  ;   0 ][ u; q ][ v; r ]+[ u; q ][ 0  ;   0 ][ v; r ]
        = qv+ ur+u r+q v
     = (γ_nq)(γ v)+ ((γ_nr)(γ u))^∗

If, in addition, (u,q)∈(A), we obtain

    A[ u; q ][ v; r ]+[ u; q ][ 0  ;   0 ][ v; r ]
        = (γ_nq)(γ v)+ ((γ_nr)(γ u))^∗
     =-ıγ uγ v_Ł(^d-1)+((γ_nr)(γ u))^∗
       = γ uıγ v_Ł(^d-1)+((γ_nr)(γ u))^∗
    =((ıγ v+γ_nr)(γ u))^∗.

Since for every u∈𝒟, we find q∈𝒟^d
such that (u,q)∈(A), 

    γ[𝒟]⊆γ u∃ q∈ H(,Ω)(u,q)∈(A).

Thus, the set on the right-hand side is dense in H^1/2(^d-1). This in
turn implies that (v,r)∈(A^*) if and only if ıγ v+γ_nr=0,
and in this case we have A^*(v,r)=-A(v,r). This implies that
A is skew-selfadjoint.



 The factor ı in front of γ u is chosen as a mere convenience in order to render the corresponding operator A in thm:Robinssa skew-selfadjoint. It is also possible to choose β∈(H^1/2(∂Ω)) with -β≥ 0 instead of ı. Then one obtains for all U∈(A) and V∈(A^*) the estimates UAU≥ 0 and VA^*V≥ 0. Appealing to rem:Aaccretive, it can be shown that the corresponding evolutionary equation
 
    (νM(ν)+A)U=F

 for a suitable material law M as in thm:Solution_theory_EE is well-posed.



Next, one could argue that in the case for arbitrary Ω, the
condition 

    ıπ_()u+_π_()q=0

amounts to a generalisation of the Robin boundary condition just
considered. However, this is not true as the following proposition shows.

 Let u∈ H^1(Ω),q∈ H(,Ω). Moreover, we set κ()→Ł(^d-1) with κ v=γ v for v∈(). Then γ_nq + ıγ u=0 if and only if
 
    _π_() q + ıκ^∗κπ_() u=0.




 We first observe that κπ_() w=γ w for each w∈ H^1(Ω).

 Assume now that γ_nq + ıγ u=0 and let v∈(). Then we compute, using prop:ibpabstract and eq:ibptrace
 
    ıκ^∗κπ_() uv_()   = ıκπ_()uκ v_Ł(^d-1)
       = ıγ uγ v_Ł(^d-1)
       = -(γ_nq)(γ v)
       = - qv_Ł(Ω)+-q v_Ł(Ω)^d
       = -_π_()qv_(),

which proves one of the asserted implications.

Assume now that _π_() q + ıκ^∗κπ_() u=0 and let v∈ H^1/2(^d-1). We take w∈ H^1(Ω) with γ w=v and compute

    (γ_nq)(v)    =  qw_Ł(Ω)+q w_Ł(Ω)^d
       = _π_() qπ_()w_()
       = -ıκ^∗κπ_() uπ_()w_()
       = -ıκπ_() uκπ_() w_Ł(^d-1)
       = -ıγ uv_Ł(^d-1),

which shows the remaining implication.













At first, we observe that ()⊆ H≅ H'⊆()',
where the last injection is given by κ' H'→()'.

 Let Ω⊆^n open. 

(a) The operator Φ()→()'
such that for all q∈() we have

    Φ(q) u↦ qu_Ł(Ω)+q u_Ł(Ω)^n

is unitary.

(b) Define F H→ H',ϕ↦(ψ↦ϕψ).
Then

    Φ^-1κ'F=_κ^*.




Easy computation.

With the latter result, we can define the abstract Robin boundary
condition as follows. 



































§ COMMENTS


The concept of abstract trace spaces has been introduced in <cit.> in order to study a multi-dimensional analogue for port-Hamiltonian systems. Also concerning differential equations at the boundary (so-called impedance type boundary conditions), the concept of abstract boundary value spaces has been employed, see <cit.>.

A comparison between abstract and classical trace spaces has been provided in  <cit.> particularly concerning H^-1/2(^d-1).  A good introduction for trace mappings for
more complicated geometries can be found e.g. in <cit.>.  The trace operator can also be suitably established for H(,Ω)-regular vector fields given that Ω is a so-called Lipschitz domain, see <cit.>.



§ EXERCISES

tocsectionExercises


Let ϕ∈(^d), f∈Ł(^d).
Show that 

    ϕ∗ f x↦∫_^dϕ(x-y)f(y)ỵ

belongs to H^1(^d) and that (ϕ∗ f)=(ϕ)∗ f.
If, in addition, f∈ H^1(^d)=(), then (ϕ∗ f)=ϕ∗ f,
where the convolution is always taken component wise.



Let Ω⊆^d be open. Let f∈Ł(Ω) and denote by f̃∈Ł(^d)
the extension of f by zero. Let v∈^d, τ>0 and
define f_τf̃(·+τ v)|_Ω.


  * 
Show that f_τ→ f in Ł(Ω) as τ→0.

  * 
Let now f∈ H^1(Ω) and  Ω+τ v⊆Ω for all τ>0. Show that
f_τ→ f in H^1(Ω) as τ→0. 




 Prove thm:divdensity.



 Let Ω⊆^d be open, M(M)⊆→(Ł(Ω)×Ł(Ω)^d) with M<ν_0 for some ν_0∈, c>0 such that for all z∈_≥ν_0 we have zM(z) ≥ c, ν≥max{ν_0,0} and ν≠ 0.
Show that there exists a unique 

    [ v; q ]∈ H_ν^1(;([ 0  ;   0 ]))

satisfying 

    (νM(ν)+[ 0  ;   0 ])[ v; q ]=[ 0; 0 ]    on Ω,
    π_()v(t)=ϕ(t)f     for all t∈,

for some bounded ϕ∈ C^∞() with infϕ>-∞
and f∈().



 Let Ω=^d-1×0. Show that
there exists a continuous linear operator E H^1(Ω)→ H^1(^d)
such that E(ϕ)|_Ω=ϕ for each ϕ∈ H^1(Ω). 


[Korn's second inequality]
 Let Ω=^d-1×0. Using exer:Extension
show that there exists c>0 such that for all ϕ∈ H^1(Ω)^d
we have

    ϕ_H^1(Ω)^d≤ c(ϕ_Ł(Ω)^d+ϕ_Ł(Ω)^d× d).

Thus, describe the space of boundary values of (). 

Hint: Prove a corresponding result for Ω=^d first
after having shown that (^d)^d forms a dense subset of
both H^1(Ω)^d and ().



 Let Ω⊆^3 be open. Compute
() H_0(,Ω)^_H(,Ω)
and show that ()→()
is well-defined, unitary and skew-selfadjoint.







[heading=subbibliography]


CHAPTER: CONTINUOUS DEPENDENCE ON THE COEFFICIENTS I


The power of the functional analytic framework for evolutionary equations
lies in its variety. In fact, as we have outlined in earlier lectures,
it is possible to formulate many differential equations in the form

    (∂_tM(∂_t)+A)U=F.

In this chapter we want to use this versatility and address continuity
of the above expression (or more precisely of the solution operator)
in M(∂_t). To see this more clearly, fix F and take a sequence
of material laws (M_n)_n. We will address the following
question: what are the conditions or notions of convergence of (M_n)_n
to some M in order that (U_n)_n with U_n given as the solution of

    (∂_tM_n(∂_t)+A)U_n=F

converges to U, which satisfies 

    (∂_tM(∂_t)+A)U=F?


In the first of two chapters on this subject, we shall specialise to A=0;
that is, we will discuss ordinary differential equations with infinite-dimensional
state space. To begin with, we address the convergence of material laws
pointwise in the Fourier–Laplace transformed domain and its relation
to the convergence of material laws evaluated at the time derivative.



§ CONVERGENCE OF MATERIAL LAWS


Throughout, let H be a Hilbert space. We briefly
recall that a sequence (T_n)_n in (H) converges in the
strong operator topology to some T∈(H) if for all
x∈ H we have

    T_nx→ Tx  (n→∞).

(T_n)_n is said to converge in the weak operator topology to T∈(H) if for all x,y∈ H we have

    yT_nx→yTx  (n→∞).

We denote the set of material laws on H with abscissa of boundedness less than or equal to
ν_0∈ by

    ℳ(H,ν_0)M(M)→(H)M material law,M≤ν_0.



 Let ν_0∈, ν>ν_0. Then ℳ(H,ν_0) is an algebra and ℳ(H,ν_0)∋ M↦ M(ν)∈((;H)) is an algebra homomorphism which is one-to-one by thm:representaion. 



Let ν_0∈.
A sequence (M_n)_n∈ in ℳ(H,ν_0) is called bounded
if 

    sup_n∈M_n_∞,_>ν_0<∞.




Let ν_0∈, (M_n)_n in ℳ(H,ν_0) be
bounded. Assume that for all z∈_>ν_0 the sequence (M_n(z))_n
converges in the weak operator topology of (H) with limit M(z) and let ν>ν_0. Then M∈ℳ(H,ν_0) and M_n(ν)→ M(ν) as
n→∞ in the weak operator topology of ((,H)).

If, in addition, (M_n(z))_n converges in the strong
operator topology of (H) for all z∈_>ν_0, then, as n→∞, M_n(ν)→ M(ν) in the strong operator topology of ((,H)).



Let z_0∈_>ν_0, r∈0 z_0-ν_0. For x,y∈ H,
by Cauchy's integral formula, we deduce

    yM_n(z_0)x=1/2πı∫_∂ B(z_0,r)yM_n(z)x_H/z-z_0ẓ  (n∈).

Using boundedness of (M_n)_n, Lebesgue's dominated
convergence theorem yields

    yM(z_0)x=1/2πı∫_∂ B(z_0,r)yM(z)x_H/z-z_0ẓ.

Since 

    yM(z)x_H≤x_Hy_Hsup_n∈M_n_∞,_>ν_0  (z∈_>ν_0),

yM(·)x_H is holomorphic in a neighbourhood of z_0.
By exer:holomorphic we obtain that M_>ν_0→(H)
is holomorphic. In fact, the estimate eq:elem_est_M
even implies that M∈ℳ(H,ν_0). 

If z∈_>ν_0 and (M_n(z))_n even converges in the strong operator topology, then the limit is clearly M(z).

The convergence statements for (M_n(ν))_n (in the weak and strong operator topology)
are then implied by Fourier–Laplace transformation. 



In thm:converge_wotsot, it suffices
to assume that (M_n(z))_n converges only for z belonging
to a countable subset of _>ν_0 with an accumulation point in
_>ν_0.


The next statement is essential for the convergence statement for
“ordinary” evolutionary equations.


Let (T_n)_n be a sequence in (H)
converging in the strong operator topology to some T∈(H) with
0∈⋂_n∈ρ(T_n), sup_n∈T_n^-1<∞
and (T)⊆ H dense. Then T is continuously invertible
and (T_n^-1)_n converges to T^-1 in the strong
operator topology.



We set Ksup_n∈T_n^-1. We show that T is
continuously invertible first. For this, let x∈ H. Then 

    x=T_n^-1T_nx≤ KT_nx→ KTx  (n→∞).

Hence, T is one-to-one and it follows that (T)⊆ H
is closed. Hence, 0∈ρ(T). For x∈ H we conclude

    T_n^-1x-T^-1x    =T_n^-1(T-T_n)T^-1x≤ K(T-T_n)T^-1x→0  (n→∞).*



We are now in the position to obtain the first result on continuous dependence.



Let ν_0∈, (M_n)_n a bounded sequence
in ℳ(H,ν_0), c>0 such that
for all n∈ and z∈_>ν_0 we have

    zM_n(z)≥ c.

If (M_n(z))_n converges in the strong operator
topology for all z∈_>ν_0 then for the limit M(z) we have M∈ℳ(H,ν_0)
with zM(z)≥ c for all z∈_>ν_0 and for ν>ν_0 we have

    (νM_n(ν))^-1→(νM(ν))^-1

in the strong operator topology.



By thm:converge_wotsot, we observe M∈ℳ(H,ν_0). Let z∈_>ν_0.
Then we have zM(z) = lim_n→∞ zM_n(z)≥ c and hence
zM(z) is continuously invertible. Since 0∈⋂_n∈ρ(zM_n(z))
and (zM_n(z))^-1≤1/c by prop:block_op_realinv<ref>, we deduce by prop:sotInv
applied to T_n=zM_n(z) that (zM_n(z))^-1→(zM(z))^-1
in the strong operator topology. By thm:converge_wotsot, for ν>ν_0
we infer (νM_n(ν))^-1→(νM(ν))^-1
in the strong operator topology.




§ A LEADING EXAMPLE


We want to illustrate the findings of the previous section with the
help of an ordinary differential equation. Also, we shall provide an argument on the limitations
of the theory presented above. Let (Ω,Σ,μ)
be a finite measure space. 

Note that for V∈ L_∞(μ) with associated multiplication operator V() as in thm:mult1 we have that

    M z↦1+z^-1V()∈(Ł(μ))

is a material law with M=0 unless V=0 (in case V=0 we have M = -∞).
The corresponding evolutionary equation is given by

    νu+V()u=f.

We want to study sequences of material laws of this form; that is, material laws induced by sequences (V_n)_n in L_∞(μ).
First, we provide the following characterisation of the convergence
of multiplication operators. 
We recall that for a Banach space X the weak^* topology σ(X',X) on X' is the coarsest topology such that all the mappings X'∋ x'↦ x'(x) (x∈ X) are continuous.


 Let (V_n)_n in L_∞(μ) and
V∈ L_∞(μ). Then the following statements hold.


  *  V_n()→ V()
in (Ł(μ)) if and only if V_n→ V in L_∞(μ).


  *  V_n()→ V() in the strong
operator topology of (Ł(μ)) if and only if (V_n) is bounded in L_∞(μ) and V_n→ V
in L_1(μ).


  *  V_n()→ V() in the weak operator topology of (Ł(μ)) if and
only if V_n→ V in the weak^* topology σ(L_∞(μ),L_1(μ)).




<ref> This is a direct consequence of prop:mult1.75.

<ref> Assume V_n→ V in L_1(μ) and that (V_n)_n
is bounded in L_∞(μ). Then (V_n-V)_n is also bounded in L_∞(μ).
For f∈ L_∞(μ)⊆Ł(μ) we obtain

    V_n()f - V()f_Ł(μ)^2     = ∫_ΩV_n - V^2 f^2μ̣
       ≤sup_n∈V_n-V_L_∞(μ)f_L_∞(μ)^2 ∫_ΩV_n-V dμ→ 0.

Since L_∞(μ) is dense in Ł(μ) and (V_n()-V())_n is bounded by prop:mult1.75, we obtain V_n()→ V() in the strong operator topology of (Ł(μ)).

Now, let V_n()→ V() in the strong operator
topology of (Ł(μ)).
Then (V_n())_n is bounded
in (Ł(μ)) by the uniform boundedness
principle. Now prop:mult1.75 yields boundedness of (V_n)_n in L_∞(μ).
Moreover, since _Ω∈Ł(μ), we deduce V_n = V_n()_Ω→ V()_Ω = V
in Ł(μ). Since Ł(μ) embeds continuously into L_1(μ) we obtain V_n→ V
in L_1(μ).

<ref> The assertion follows easily upon realising that ϕ∈ L_1(μ)
if and only if there exists ψ_1,ψ_2∈Ł(μ) such that
ϕ=ψ_1ψ_2.


With the latter result at hand together with the results in the previous
section, we easily deduce the next theorem on continuous dependence
on the coefficients.


Let (V_n)_n in L_∞(μ) be bounded, V∈ L_∞(μ), and V_n→ V in L_1(μ). Then there
exists ν>0 such that 

    (ν+V_n())^-1→(ν+V())^-1

in the strong operator topology of ((;Ł(μ))).


Note that the convergence statement
can be improved, see exer:improved_cd_sot_ex.


  By prop:conve_mult<ref> we obtain V_n()→ V() in the strong
operator topology of (Ł(μ)). Note that for ν≥ 1+sup_n∈V_n_L_∞(μ) we have

    (z+V_n()) ≥ 1    (z∈_>ν, n∈).

Now thm:ODEcondep applied to M_n(z) = 1+z^-1V_n() yields the assertion.



  thm:cd_sot_ex can be generalized in the following way. Let (B_n)_n in (H), B∈(H), B_n→ B in the strong operator topology.
  Then there txists ν>0 such that 

    (ν+B_n)^-1→(ν+B)^-1

in the strong operator topology of ((;Ł(μ))).


In thm:cd_sot_ex we did assume strong convergence of the sequence of multiplication operators (V_n())_n.
A natural question to ask is whether the stated result can be improved to (V_n)_n
converging in the weak^* topology σ(L_∞(μ),L_1(μ)) only. The answer is neither
`yes' nor `no', but rather `not quite', as we will show in the following.
We start with a result on weak^* limits of scaled periodic functions, which will serve as the prototypical example for a sequence converging in the weak^* topology of L_∞.


 Let f∈ L_∞(^d) be 01^d-periodic; that is,

    f(·+k)=f  (k∈^d).

Then 

    f(n·)→∫_01^df(x)x̣_^d

in the weak^* topology σ(L_∞(^d),L_1(^d)) as n→∞.



Without loss of generality, we may assume ∫_01^df(x)x̣=0.
By the density of simple functions in L_1(^d) and the boundedness
of (f(n·))_n in L_∞(^d), it suffices to show

    ∫_Qf(nx)x̣→0  (n→∞)

for Q=aba_1b_1×…×a_db_d where
a=(a_1,…,a_d),b=(b_1,…,b_d)∈^d. 
By translation and the periodicity of f we may assume a=0. Thus, it suffices to show

    ∫_0bf(nx)x̣→0  (n→∞)

for all b∈0∞^d. So, let b=(b_1,…,b_d)∈0∞^d.
Let n∈. Then we find z∈^d and ζ∈01^d
such that nb=z+ζ.
We compute

    ∫_0bf(nx)x̣
        =1/n^d∫_0nb f(x)x̣
        =1/n^d∫_0z_1×0nb_2×…×0nb_df(x)x̣
     +1/n^d∫_z_1z_1+ζ_1×0nb_2×…×0nb_df(x)x̣.

We now estimate

    1/n^d∫_z_1z_1+ζ_1×0nb_2×…×0nb_df(x)x̣   ≤1/n^d∫_z_1z_1+ζ_1×0nb_2×…×0nb_df(x)x̣
       ≤1/n^d∫_01×0nb_2×…×0nb_dx̣f_L_∞(μ)
        =1/nb_2·…· b_df_L_∞(μ).

Continuing in this manner and using z_j≤ nb_j for all j∈{1,…,d}, we obtain

    ∫_0bf(nx)x̣   ≤1/n^d∫_0zf(x)x̣+1/n∑_j=1^db_1·…· b_d/b_jf_L_∞(μ).

Since f is 01^d-periodic and z∈^d we observe

    ∫_0zf(x)x̣ = ∏_j=1^d z_j ∫_01^d f(x)x̣ = 0.

Thus,

    ∫_0bf(nx)x̣≤1/n∑_j=1^db_1·…· b_d/b_jf_L_∞(μ),

which tends to 0 as n→∞.



Note that thm:periodic_homo also yields 

    f(n·)→∫_[0,1)^df(x)x̣_Ω

in the weak^* topology σ(L_∞(Ω),L_1(Ω)) for all
measurable subsets Ω⊆^d with non-zero Lebesgue measure.


We now present an example which shows that weak^* convergence of (V_n)_n does not yield the result of thm:cd_sot_ex.



Let (Ω,Σ,μ)=(01,ℬ(01),λ|_01).
For n∈ let V_n be given by V_n(x)sin(2π nx)
for x∈01. Then, by thm:periodic_homo, we obtain V_n→0 in σ(L_∞(01),L_1(01)) as n→∞.
Let ν>1. Then (ν+V_n()) is continuously
invertible as an operator in (;Ł(01)). Let f̃∈ C(01)
and denote f(t,x)↦_0(t)f̃(x). Then f∈(;Ł(01)).
The solution u_n∈(;Ł(01)) of 

    (ν+V_n())u_n=f

is given by the variations of constants formula; that is,

    u_n(t,x)=_0∞(t)∫_0^texp(-(t-s)sin(2π nx))ṣf̃(x)  (t∈,x∈01).

Thus, if a variant of thm:cd_sot_ex were true also in
this case, (u_n)_n needs to converge (in some sense) to the solution u
of

    νu=f,

which is given by

    u(t,x)=_0∞(t)tf̃(x)  (t∈,x∈01).

However, by thm:periodic_homo, for x∈01 we deduce 

    ∫_0^texp(-(t-s)sin(2π nx))ṣ→∫_0^tJ(-(t-s))ṣ  (n→∞)

in σ(L_∞(01),L_1(01)) for each t≥0, where

    J(s)∫_0^1exp(ssin(2π x))x̣  (s∈)

denotes the 0-th order modified Bessel function of the first kind,
cf. <cit.>. Moreover, for φ∈(), A∈ℬ(01) and using domianted convergence we obtain

    u_nφ_A_(;Ł(01))
        =∫_0^∞∫_0^1∫_0^texp(-(t-s)sin(2π nx))ṣf̃(x)_A(x)x̣φ(t)^-2ν tṭ
       →∫_0^∞∫_0^1∫_0^tJ(-(t-s))ṣf̃(x)_A(x)x̣φ(t)^-2ν tṭ
        =ũφ_A_(;Ł(01))

with 

    ũ(t,x)_0∞(t)∫_0^tJ(-(t-s))ṣf̃(x)  (t∈,x∈01).

Since (u_n)_n is bounded in (;Ł(01)) and φ_AA∈ℬ(01), φ∈()
is total in (;Ł(01)) by lem:dense sets 2,
we infer u_n→ũ
weakly in (;Ł(01)) as n→∞. In particular,
ũ u.
Furthermore, ũ is not of the form

    ∫_0^texp(-(t-s)Ṽ(x))ṣf̃(x)

for some Ṽ∈ L_∞(01) and hence, we cannot
hope for ũ to satisfy an equation of the type

    (ν+Ṽ())ũ=f.



As we shall see next, in the framework of evolutionary equations it is possible to
derive an equation involving suitable limits of (V_n)_n
and f as a right-hand side.



§ CONVERGENCE IN THE WEAK OPERATOR TOPOLOGY


In this section, we consider a particular class of material laws and characterise convercenge of the solution operators of the corresponding evolutionary equations in the weak operator topology. The main theorem that will
serve to compute the limit equation satisfied by ũ in exa:sin_Memory
reads as follows.

Let H be a Hilbert space, (B_n)_n a bounded sequence in (H) and ν>sup_n∈B_n.
Then ((ν+B_n)^-1)_n converges
in the weak operator topology of ((;H)) if and only if
for all k∈ the sequence (B_n^k)_n converges
in the weak operator topology of (H). In either case, we have 

    (ν+B_n)^-1→∑_k=0^∞(-ν^-1)^kC_kν^-1

in the weak operator topology of ((;H)), where C_k∈(H) denotes the weak limit of (B_n^k)_n for k∈ and C_0 1_H.




In the situation of thm:char_thm_wot, let B_n^k→ C_k in the weak operator topology for all k∈.
Let Lsup_n∈B_n, ν>2L, and f∈(;H).
By thm:char_thm_wot, if (ν+B_n)u_n=f for all n∈, then
(u_n)_n converges weakly in (;H) to some element ũ∈(;H).
In order to determine the differential equation satisfied by ũ,
we make the following observations: by weak convergence,

    C_k≤lim inf_n→∞B_n^k≤ L^k.

Hence, since ν^-1_≤1/ν (see sec:time_derivative) we infer that

    ∑_k=1^∞(-ν^-1)^kC_k

converges in ((;H)) and 

    ∑_k=1^∞(-ν^-1)^kC_k≤∑_k=1^∞ν^-1^k C_k < ∑_k=1^∞1/2^k = 1.

Hence, since C_0=_H we deduce that ∑_k=0^∞(-∂_t,ν^-1)^kC_k
is boundedly invertible by the Neumann series. Thus, we obtain

    f     = ν(∑_k=0^∞(-ν^-1)^kC_k)^-1ũ
     =ν(_H+∑_k=1^∞(-ν^-1)^kC_k)^-1ũ
        =ν∑_ℓ=0^∞(-∑_k=1^∞(-ν^-1)^kC_k)^ℓũ
     =νũ+ν∑_ℓ=1^∞(-∑_k=1^∞(-ν^-1)^kC_k)^ℓũ.



Before we prove thm:char_thm_wot we revisit exa:sin_Memory.

[exa:sin_Memory continued]
By thm:char_thm_wot, we need to compute the limit
of (sin^k(2π n·))_n in the weak^* topology
of L_∞(01) for all k∈. By thm:periodic_homo, we
obtain for all k∈  

    lim_n→∞sin^k(2π n·)     =∫_0^1sin^k(2πξ)ξ̣_01
     =(2m)!/(m!2^m)^2_01,     k=2m for some m∈,
    
    0,     k odd,

in σ(L_∞(01),L_1(01)). Hence, u_n→ũ weakly,
where ũ satisfies

    νũ+ν∑_ℓ=1^∞(-∑_m=1^∞ν^-2m(2m)!/(m!2^m)^2)^ℓũ=f

for ν>2 by rem:char_thm_wot.



 Before we prove the equivalence, we make some observations. Since
ν>sup_n∈B_n L, by a Neumann series argument we deduce that

    (ν+B_n)^-1=∑_k=0^∞(-ν^-1B_n)^kν^-1 = ∑_k=0^∞(-ν^-1)^k B_n^k ν^-1.

The series ∑_k=0^∞(-ν^-1)^kB_n^kν^-1
is absolutely convergent in ((;H)). Also note that for M_n_>L∋ z↦∑_k=0^∞(-1/z)^k B_n^k1/z we have M_n∈ℳ(H,ν). 

Assume now that (B_n^k)_n converges in the weak operator
topology to some C_k for all k∈. A little computation reveals that 
as n→∞, 

    M_n(z)→∑_k=0^∞(-1/z)^kC_k1/z M(z)  (z∈_>L)

in the weak operator topology, where the series on the right-hand
side converges in (H) since 

    C_k≤lim inf_n→∞B_n^k≤ L^k  (k∈).

Moreover, since ν>L, the sequence (M_n)_n is bounded in
ℳ(H,ν) and thus, M∈ℳ(H,ν) and 

    M_n(ν)→ M(ν)

in the weak operator topology by thm:converge_wotsot.

Now, we assume that ((ν+B_n)^-1)_n converges in the weak operator topology.
Then (M_n(ν))_n converges in the weak operator topology. 
Let k∈. We need to show that for all ϕ,ψ∈ H the sequence
(ϕB_n^kψ_H)_n is convergent to
some number c_k,ϕ,ψ as n→∞.
The Riesz representation theorem then yields the existence of C_k∈(H)
with ϕC_kψ=c_k,ϕ,ψ. So, let ϕ,ψ∈ H.
Moreover, we consider the functions m_n and h_n given by

    m_n(z)∑_k=0^∞(-z)^kzϕB_n^kψ_H  (z∈ B(0,1/L), n∈)

and 

    h_n(z)ϕM_n(z)ψ_H=∑_k=0^∞1/z(-1/z)^kϕB_n^kψ_H  (z∈_>L, n∈).

Clearly, m_n and h_n are holomorphic on their respective domains
for each n∈ and the sequences (m_n)_n and (h_n)_n are
uniformly bounded on compact subsets (in other words they form normal
families). Moreover, 

    m_n(z)=h_n(1/z)  (z∈ B(1/(2L),1/(2L)),n∈).

We aim to show that the coefficients of the power series of m_n
converge as n tends to infinity. The proof will be done in two
steps. In step 1, we will prove that the sequence (h_n)_n
converges to a holomorphic function h_>L→ uniformly on
compact sets. Then, in the second step, we will use this to deduce
that (m_n)_n also converges uniformly on compact sets and prove
the assertion with the help of Cauchy's integral formula.

Step 1: By prop:mat_law_function_of_td, (M_n(ı+ν))_n
converges in the weak operator topology of (Ł(;H)). For f,g∈Ł()
we thus obtain that

    (fh_n(ı+ν)g_Ł())_n     =(fϕM_n(ı+ν)gψ_Ł(;H))_n

is convergent. Thus, using Ł()·Ł()=L_1(), we obtain
that

    Ψ L_1()∋ u↦lim_n→∞(∫_h_n(ı t+ν)u(t)ṭ) ∈

defines a linear functional, which is continuous, since 

    sup_n∈sup_t∈M_n(ı t+ν)_(H)=sup_n∈M_n(ı+ν)_(Ł(;H))<∞

by boundedness of (B_n)_n. Hence, since L_1()'=L_∞(),
we find a unique h̃∈ L_∞() with

    lim_n→∞∫_h_n(ı t+ν)u(t)ṭ    =∫_h̃(t)u(t)ṭ  (u∈ L_1()).

We now show that every subsequence (h_n_k)_k of (h_n)_n has a subsequence (h_n_k_l)_l which converges locally uniformly to a holomorphic function h_>L→ such that h(ı·+ν) = h̃ a.e., and that this implies that the limit h does not depend on the subsequences.
Then we conclude that (h_n)_n itself converges locally uniformly to h.

So, let (h_n_k)_k be a subsequence of (h_n). 
By Montel's theorem (see <cit.>), we find a subsequence
(h_n_k_l)_l of (h_n_k)_k such that h_n_k_l→ h
as l→∞ uniformly on compact subsets of _>L
for some holomorphic function h_>L→. In particular, we
obtain

    lim_k→∞∫_h_n_k_l(ı t+ν)φ(t)ṭ=∫_h(ı t+ν)φ(t)ṭ  (φ∈())

by dominated convergence and hence, h(ı t+ν)=h̃(t) for
almost every t∈. This shows that the limit h is independent of choice of the subsequences (h_n_k)_k and (h_n_k_l)_l.
Indeed, if ĥ_>L→ is the limit of another subsubsequence of (h_n)_n as above, then ĥ(ı·+ν) = h̃=h(ı·+ν) a.e.
Since ĥ and h are holomorphic, the identity theorem yields ĥ=h.

Now, assume for a contradiction that (h_n)_n does not converge locally uniformly to h. Then we
find a subsequence (h_n_k)_k of (h_n)_n, a compact set K⊆_>L
and ε>0 such that

    h_n_k-h_∞,K≥ε  (k∈).

However, the subsequence (h_n_k)_k has a subsequence (h_n_k_l)_l which converges locally uniformly to h, contradicting eq:subsubsequence.
Thus, (h_n)_n itself converges locally uniformly to h, and, in particular, h_n→ h pointwise on _>L.


Step 2: By what we have shown in Step 1, the sequence (m_n)_n∈
converges pointwise on B(1/(2L),1/(2L)). Since (m_n)_n
is also uniformly bounded on compact subsets of B(0,1/L), we derive
that (m_n)_n converges uniformly on compact subsets of B(0,1/L)
by Vitali's theorem (see <cit.>). Choosing
0<r<1/L, we thus obtain by Cauchy's integral formula 

    ϕB_n^kψ_H=(-1)^k1/2πı∫_∂ B(0,r)m_n(z)/z^k+2ẓ.

Thus (B_n^k)_n converges in the weak operator topology as n→∞.























§ COMMENTS


The problems discussed here are contained in <cit.> for
both the weak and the strong operator topology. The case of differential-algebraic
equations has been invoked as well. 

The appearance of memory effects; that is, the occurence of higher
order integral operators due to a weak convergence of the coefficients
has been first observed by Tartar and can, for instance, be found in
<cit.>. The limit equation, however, is described by a
convolution term rather than a power series of integral operators.
It is, however, possible to reformulate these resulting equations into
one another, see <cit.>. 

The last characterisation of weak convergence in thm:char_thm_wot was formulated
for the first time in <cit.>.



§ EXERCISES

tocsectionExercises


Let (V_n)_n in L_∞(^d) and V∈ L_∞(^d). Characterise
convergence of V_n()→ V() in the strong operator topology
of (Ł(^d)) in terms of convergence of (V_n)_n similar
to as was done in prop:conve_mult.



Show that there exists an unbounded sequence (V_n)_n in L_∞(01)
and V∈ L_∞(01) with V_n→ V in L_1(01). 




Let (Ω,Σ,μ) be a finite measure space, (V_n)_n a
bounded sequence in L_∞(μ) and assume that V_n→ V
in L_1(μ) for some V∈ L_∞(μ). Show that there
exists ν>0 such that 

    (ν+V_n())^-1→(ν+V())^-1

in the strong operator topology of ((;Ł(μ)),H_ν^1(;Ł(μ))).



Let D=⋃_n∈n+1/2n+1, V_n_D(n·).
For suitable ν>0 compute the limit of 

    ((ν+V_n())^-1)_n

in the weak operator topology of (;Ł(01)).



Let H be a Hilbert space, c>0 and c≤ B_n=B_n^*∈(H)
for all n∈. Characterise, in terms of convergence
of (B_n)_n in a suitable sense, that

    ((νB_n)^-1)_n

converges in the weak operator topology. In the case of convergence, find
its limit and a sufficient condition for which there exists a B∈(H)
such that 

    (νB_n)^-1→(νB)^-1

in the weak operator topology.



Let H be a Hilbert space. Show that B_(H)B∈(H)B≤1
is a compact subset under the weak operator topology. If, in addition,
H is separable, show that B_(H) is also metrisable under
the weak operator topology.



Let H be a separable Hilbert space, (B_n)_n
in (H) bounded. Show that there exists a subsequence (B_n_k)_k of (B_n)_n, a material law M(M)→(H) and ν>0 such that
given f∈(;H) and (u_k)_k in (;H) with 

    νu_k+B_n_ku_k=f  (k∈),

we deduce that (u_k)_k converges weakly to some
u∈(;H) with the property that

    νM(ν)u=f.




[heading=subbibliography]



CHAPTER: CONTINUOUS DEPENDENCE ON THE COEFFICIENTS II


This chapter is concerned with the study of problems of the form

    (νM_n(ν)+A)U_n=F

for a suitable sequence of material laws (M_n)_n
when A≠0. The aim of this chapter will be to provide the conditions required for
convergence of the material law sequence to imply the existence
of a limit material law M such that the limit U=lim_n→∞U_n
exists and satisfies 

    (νM(ν)+A)U=F.

Additionally, for material laws of the form
M_n(ν)=M_0,n+ν^-1M_1,n it will be desirable to have the respective limit
material law satisfy M(ν)=M_0+ν^-1M_1 for
some M_0,M_1∈(H). This cannot be expected (as we have seen in the guiding example in the
previous chapter) if A is
a bounded operator, the Hilbert space H is infinite-dimensional, and
the material law sequence only converges pointwise in the weak operator
topology. It will turn out, however, that if A is “strictly unbounded”
then a suitable result can hold, even if we only assume weak convergence
of the material law operators. 



§ A CONVERGENCE THEOREM


The main convergence theorem of this chapter will be presented next.

Let H be a Hilbert space, ν_0∈, (M_n)_n
in ℳ(H,ν_0) and M∈ℳ(H,ν_0). Assume there
exists c>0 such that for all n∈ we have

    zM_n(z)≥ c  (z∈_>ν_0).

Let A(A)⊆ H→ H be skew-selfadjoint and assume (A)↪ H compactly. If M_n(z)→ M(z)
as n→∞ in the weak operator topology for all z∈_>ν_0,
then 

    (νM_n(ν)+A)^-1→(νM(ν)+A)^-1

in the strong operator topology of ((;H)) for each ν>ν_0. 


For the proof of this theorem, we need a lemma first.

Let H be a Hilbert space, A(A)⊆ H→ H
skew-selfadjoint, c>0, (T_n)_n in (H) with T_n≥ c for all n∈, and T∈(H). 
Assume (A)↪ H compactly and T_n→ T in the
weak operator topology. Then 0∈⋂_n∈ρ(T_n+A)∩ρ(T+A)
and 

    (T_n+A)^-1→(T+A)^-1

in the norm topology of (H). 



From T_n≥ c it follows that 0∈ρ(T_n+A) (n∈) and ((T_n+A)^-1)_n
is bounded in (H). Indeed, since B T_n+A satisfies B= T_n≥ c and (B)=(A)=(B^∗) due to the skew-selfadjointness of A, prop:accr_inv yields the assertion. Moreover, since

    A(T_n+A)^-1=1-T_n(T_n+A)^-1

for all n∈, it follows that ((T_n+A)^-1)_n is also bounded in (H,(A)) by the boundedness of (T_n)_n in (H). Due to
the convergence of (T_n)_n to T, it follows that T≥ c, and thus, (T+A)^-1∈(H,(A)).
Before we come to a proof of the desired result, we will prove an
auxiliary observation.

Claim: for all (f_n)_n in H weakly converging to f, we
have (T_n+A)^-1f_n→(T+A)^-1f
in the norm topology of H. 

For proving the claim, let (f_n)_n in H be weakly convergent
to some f. Consider u_n (T_n+A)^-1f_n. Then (u_n)_n is bounded in (A), since ((T_n+A)^-1)_n is bounded in (H,(A)) and (f_n)_n is bounded in H. Hence, there exists a subsequence (u_n_k)_k which weakly converges to some u in (A).
Since (A)↪ H compactly, we infer u_n_k→ u in
the norm topology of H. Hence, in the
equality

    T_n_ku_n_k+Au_n_k=f_n_k,

as T_n_k→ T in the weak operator topology and u_n_k→ u in H, we may let k→∞ and obtain for the weak limits

    Tu+Au=f;

that is, u=(T+A)^-1f. Having identified the limit,
a contradiction argument (here a so-called `subsequence argument', see exer:convPDEstrong) concludes that (u_n)_n itself converges
weakly in (A) and strongly in H to u. Thus, the claim
is proved.

Next, assume by contradiction that ((T_n+A)^-1)_n
does not converge in operator norm to (T+A)^-1. Then
we find an ε>0 and a strictly increasing sequence of
integers, (n_k)_k, and a sequence of unit vectors (f_n_k)_k
in H such that 

    (T_n_k+A)^-1f_n_k-(T+A)^-1f_n_k≥ε.

By possibly taking another subsequence, we may assume without loss
of generality that (f_n_k)_k converges weakly
to some f∈ H. By the claim proved above, we deduce  (T_n_k+A)^-1f_n_k→(T+A)^-1f
and (T+A)^-1f_n_k→(T+A)^-1f, both
in the norm topology of H as k→∞. Thus, we may let k→∞
in eq:contr, and obtain the desired contradiction.



 By thm:converge_wotsot it suffices to show that for
all z∈_>ν_0 

    (zM_n(z)+A)^-1→(zM(z)+A)^-1  (n→∞)

 in the strong operator topology. This, however, follows from lem:convPDE
applied to T_n=zM_n(z).


Note that we only used convergence in the strong operator topology
in the proof of thm:convPDE. However, the assertion in
lem:convPDE is about convergence in the norm topology.
The reason that we cannot assert the convergence claimed in thm:convPDE
in the norm topology is that the compact embedding of (A)↪ H
only works locally for fixed z, and not uniformly in z. This
situation can, however, be rectified. We refer to exer:normPDEconv
for this.





§ THE THEOREM OF RELLICH AND KONDRACHOV


In order to apply thm:convPDE, we need to provide a setting
where the condition on the compactness of the embedding is satisfied.
In fact, it is true that H^1(Ω) embeds compactly into Ł(Ω)
given Ω⊆^d is bounded and has `continuous boundary', see e.g. <cit.>. In this
chapter, we restrict ourselves to a proof of a less general statement. 

A preparatory result needed to prove the compact embedding theorem is given
next.

 Let I⊆ be an open, bounded, non-empty
interval. Then the mapping H^1()∋ f↦ f|_I∈ H^1(I)
is well-defined, continuous and onto. Moreover, there exists a continuous right inverse H^1(I)→ H^1().


For the proof of this proposition, we need an auxiliary result first.


 Let Ω⊆^d be open and connected. Moreover, let u∈ H^1(Ω) with u=0. Then u is constant.


We leave the proof of this lemma as exer:kernel_grad.


The mapping H^1()→ H^1(I),f↦ f|_I
is readily confirmed to be continuous. It remains to prove that it is onto. Let I=ab, u∈ H^1(I) and define the function v by

    v(t)∫_a^t ∂ u(s)ṣ   (t∈ab).

Clearly, v∈Ł(ab) and we compute for each φ∈(ab)

    vφ'_Ł(ab)    = ∫_a^b (∫_a^t ∂ u(s)ṣ)^∗φ'(t) ṭ
     = ∫_a^b ∫_s^b φ'(t)ṭ ∂ u(s)^∗ṣ
        = -∂ uφ_Ł(ab).

This shows v∈ H^1(ab) with ∂ v=∂ u. Hence, by lem:kernel_grad there exists a constant c∈ with u=c+v. We now define f by

    f(t) 
                     0     t<a-1  t>b+1,
    
                     ct+c(1-a)      a-1≤ t ≤ a,
    
                     u(t)     a<t<b,
    
                     -(c+v(b))t+(c+v(b))(1+b)     b≤ t≤ b+1.

We then easily see that f∈ H^1() and clearly f|_ab=u. In order to see that u↦ f is continuous, we need to establish that the value c depends continuously on u. This, however, follows from the estimate

    |c|   =1/√(b-a)(∫_a^b |c|^2)^1/2≤1/√(b-a) (u_Ł(a,b)+v_Ł(a,b)) 
       ≤1/√(b-a)(u_Ł(a,b) + (b-a)∂ u_Ł(a,b)) 
       ≤√(2)max{1,(b-a)}/√(b-a)u_H^1(a,b).




Let I⊆ be an open bounded interval.
Then H^1(I)↪Ł(I) compactly.


 By prop:restriction, we find a continuous mapping E H^1(I)→ H^1() such that for all u∈ H^1(I) we have E(u)|_I=u. Moreover, by exer:12Hoelder the mapping H^1()↪ C^1/2() is continuous. Thus,

    H^1(I) E→ H^1() ↪ C^1/2() → C^1/2(I),
 
is a composition of continuous mappings, where the last mapping is the restriction to I. Since C^1/2(I)↪ C(I) compactly by the Arzelà–Ascoli theorem, and C(I)↪Ł(I) continuously, we infer H^1(I)↪Ł(I) compactly.

We now have the opportunity to study the limit behaviour of a periodic
mixed type problem.

[Highly oscillatory problems]
Let s_1,s_2→01 be 1-periodic, measurable
functions. Then for ν>0, we set

    S^(n)(ν[ s_1(n)      0;      0 s_2(n) ]+[ 1-s_1(n)        0;        0 1-s_2(n) ]+[   0   ∂; ∂_0   0 ])^-1,

where ∂= and ∂_0=_0 are regarded as operators
in Ł(01) with respective domains H^1(01) and H_0^1(01).
Then, by thm:compH1, the operator A[   0   ∂; ∂_0   0 ] satisfies the assumptions of thm:convPDE. Moreover,
thm:periodic_homo implies that the remaining assumptions
of thm:convPDE are satisfied. Hence, we deduce that (S^(n))_n
converges in the strong operator topology of (;Ł(01))
to the limit

    (ν[ ∫_0^1s_1        0;        0 ∫_0^1s_2 ]+[ 1-∫_0^1s_1          0;          0 1-∫_0^1s_2 ]+[   0   ∂; ∂_0   0 ])^-1.



Next, we aim to provide an application to more than one spatial dimension.
For this, we will also need a corresponding compactness statement. This is the subject of the rest of this section.

 Let Ω⊆^d be open and bounded.
Then H_0^1(Ω)↪Ł(Ω) compactly.



Without loss of generality (by shifting and
shrinking of Ω and extending by 0), we may assume that Ω=01^d.
We carry out the proof by induction on the spatial dimension d.
The case d=1 has been dealt with in thm:compH1. Assume
the statement is true for some d-1. Using that (01^d) is dense in H_0^1(01^d), we infer the continuity of the
injection 

    R H_0^1(01^d)    → H^1(;Ł(01^d-1))∩Ł(;H_0^1(01^d-1))
    ϕ   ↦(t↦(ω↦ϕ(t,ω))),

where we identify ϕ with its extension to ^d by 0. The range space is endowed with the usual sum scalar
product.

Let (ϕ_n)_n be a weakly convergent null-sequence
in H_0^1(01^d). In particular, (Rϕ_n)_n is bounded in H^1(;Ł(01^d-1)) and hence, it 
is also bounded in (;Ł01^d-1) by thm:Sobolev_emb (and cor:vanish_at_inf);
that is,

    sup_t∈01,n∈ϕ_n(t,·)_Ł(01^d-1)<∞.

Let f∈Ł(01^d-1). Then (ϕ_n,f)_n given by

    ϕ_n,f t↦ϕ_n(t,·)f_Ł(01^d-1)

is a weakly convergent null-sequence in H^1(01). We obtain
by thm:compH1 that ϕ_n,f→ 0 in Ł(01)
as n→∞. By separability of Ł(01^d-1) we find D⊆Ł(01^d-1) countable
and dense, a subsequence (again labeled by n)  and 
a null-set N⊆ such that ϕ_n,f(t)→ 0
for all t∈∖ N and f∈ D as n→∞. By eq:boundRellich, we deduce ϕ_n,f(t)→0
for all t∈∖ N and f∈Ł(01^d-1) as n→∞, or, in other words, ϕ_n(t,·)→ 0 weakly in Ł(01^d-1) for each t∈∖ N as n→∞.

Next, we show that there exists a null set N⊆ N_1⊆ such that ϕ_n(t,·)→ 0 in Ł(01^d-1) for all t∈∖ N_1.
For this, since (Rϕ_n)_n in Ł(;H_0^1(01^d-1)) is bounded, we find a null set N⊆ N_1⊆ such
that (ϕ_n(t,·))_n is bounded in H_0^1(01^d-1)
for all t∈∖ N_1. Let t∈∖ N_1. Then there exists a further subsequence (ϕ_n_k(t,·))_k which converges weakly in H_0^1(01^d-1). By the induction hypothesis, (ϕ_n_k(t,·))_n_k converges strongly in Ł(01^d-1), and since we have already seen that it is a weak null-sequence in Ł(01^d-1), we derive
ϕ_n_k(t,·)→ 0 in Ł(01^d-1). By a subsequence argument we derive that

    ϕ_n(t,·)→ 0

in Ł(01^d-1) for all t∈∖ N_1.

Now, for
n∈ we deduce

    ϕ_n_Ł(01^d)^2=∫_0^1ϕ_n(t,·)_Ł(01^d-1)^2ṭ→0,

where we have used dominated convergence, which is possible due to eq:boundRellich.




§ THE PERIODIC GRADIENT


In this section we investigate the gradient on periodic functions on ^d. Throughout, we set
Y01^d.
[Periodic Gradient]
We define 

    C_♯^∞(Y)ϕ|_Yϕ∈ C^∞(^d), ϕ(·+k)=ϕ (k∈^d)

and

    _♯,∞ C_♯^∞(Y)⊆Ł(Y)    →Ł(Y)^d
    ϕ   ↦ϕ.

Moreover, we set _♯-_♯,∞^* and
_♯-_♯^*=_♯,∞.


The operators just introduced can easily be shown to lie between the
operator realisations we have introduced in earlier chapters. Indeed, it is easy
to see that 

    _0⊆_♯ and _0⊆_♯

and, consequently, we also have

    _♯⊆ and _♯⊆.

The corresponding domains for the operators _♯ and
_♯ will be denoted by H_♯^1(Y)
and H_♯(,Y), respectively.

For the next results, we define the periodic extension operator. For
ϕ∈Ł(Y)^m we put

    ϕ_(x+k)ϕ(x)

for almost every x∈ Y and all k∈^d.


We start with the following two observations.


 Let f∈Ł(Y) and (ρ_k)_k be a δ-sequence in (^d) (cf. exer:delta-seq). Define
 
    f_k  (ρ_k∗ f_)|_Y    (k∈).

 Then f_k∈ C_♯^∞(Y) for each k∈ and f_k→ f in Ł(Y) as k→∞.



 It follows as in exer:C_cinfty dense that ρ_k∗ f_ is in C^∞. Moreover, one easily sees that ρ_k∗ f_ is 01^d-periodic, and hence, f_k∈ C^∞_♯(Y) for each k∈. For the convergence we observe 
 
    (ρ_k∗ (_Y+B(0,1) f_))(x)=f_k(x)   (x∈ Y, k∈).

 Moreover, by exer:C_cinfty dense we have ρ_k∗ (_Y+B(0,1) f_) →_Y+B(0,1) f_ in Ł(^d) as k→∞, and thus,
 
    f_k =(ρ_k∗ (_Y+B(0,1) f_))|_Y→ (_Y+B(0,1) f_)|_Y=f   (k→∞)   in Ł(Y).





 C_♯^∞(Y)^d is a core for _♯ and _♯Ψ=Ψ for each Ψ∈ C_♯^∞(Y)^d.



 First we note that C_♯^∞(Y)^d⊆(_♯) and _♯Ψ=Ψ for Ψ∈ C_♯^∞(Y)^d. To see this, for ϕ∈ C_♯^∞(Y),Ψ∈ C_♯^∞(Y)^d we compute
 
    ϕΨ_Ł(Y)^d    =∫_Y ϕ(x)Ψ(x)_^dx̣=-∫_Y ϕ(x)^∗Ψ (x) x̣
        =ϕ-Ψ_Ł(Y)

by integration by parts (note that the boundary values cancel out due to the periodicity of ϕ and Ψ). Now, let q∈(_♯) and (ρ_k)_k be a δ-sequence in (^d). For k∈ we define 

    q_k (ρ_k ∗ q_)|_Y,

and obtain q_k ∈ C^∞_♯(Y) and q_k→ q in Ł(Y)^d as k→∞ by lem:delta-seq-periodic. It is left to show that q_k→_♯ q in Ł(Y) as k→∞. For doing so, we show that q_k =(ρ_k ∗ (_♯ q)_)|_Y, which would then yield the assertion again by lem:delta-seq-periodic. So, let k∈ and ϕ∈ C^∞_♯(Y). We compute

    q_kϕ_Ł(Y)^d   = ∫_Y ∫_^dρ_k(y) q_ (x-y)ỵϕ(x)_^dx̣
       = ∫_^dρ_k(y)  ∫_Y q_(x-y)ϕ(x)_^dx̣ỵ
       = ∫_^dρ_k(y)  ∫_Y-yq_(x)(ϕ)_ (x+y)_^dx̣ỵ
       = ∫_^dρ_k(y)  ∫_Yq(x)(ϕ)_ (x+y)_^dx̣ỵ
       =∫_^dρ_k(y)  ∫_Yq(x)(ϕ_(· +y)) (x)_^dx̣ỵ
       = -∫_^dρ_k(y)  ∫_Y_♯ q(x)ϕ_(x+y)_^dx̣ỵ
       = -∫_^dρ_k(y) ∫_Y+y(_♯ q)_(x-y)ϕ_(x)_^dx̣ỵ
       = -(ρ_k ∗ (_♯ q)_)|_Yϕ_Ł(Y),

where we have used perodicity 
as well as ϕ_(· +y)∈ C_♯^∞(Y). 



  The proof of lem:C_sharp_core reveals that every q∈(_♯) can be approximated by elements in C^∞_♯(Y)^d∩(_♯).





Let Ω⊆^d be open,
bounded, u∈ H_♯^1(Y) and q∈ H_♯(,Y).
Then u_|_Ω∈ H^1(Ω), q_|_Ω∈ H(,Ω) and 

    (u_|_Ω)=(_♯u)_|_Ω(q_|_Ω)=(_♯q)_|_Ω.




Let first ϕ∈ C^∞_♯(Y). Then by definition ϕ_∈ C^∞(^d) and we easily see

    ϕ_ = (ϕ)_ = (_♯ϕ)_.

Moreover, since Ω is bounded, we infer ϕ_∈ H^1(Ω). By definition of H^1_♯(Y) we find a sequence (ϕ_k)_k∈ in C^∞_♯(Y) such that ϕ_k→ u in Ł(Y) and _♯ϕ_k→_♯ u in Ł(Y)^d as k→∞. Since 

    Ł(Y)→Ł(Ω),   f↦ f_

is bounded due to the boundedness of Ω, we also derive ϕ_k,→ u_ in Ł(Ω) and (_♯ϕ_k)_→ (_♯ u)_ in Ł(Ω)^d as k→∞.  By what we have shown above, we infer

    ϕ_k,=(_♯ϕ_k)_→ (_♯ u)_   (k→∞)

in Ł(Ω)^d, and thus, u_∈ H^1(Ω) with u_=(_♯ u)_ by the closedness of . The proof for q follows by the same argument with lem:C_sharp_core as an additional resource.



The extension result just established yields the following compactness
statement.

 The embedding H_♯^1(Y)↪Ł(Y)
is compact.



Let (ϕ_n)_n be a bounded sequence in H_♯^1(Y).
Let Ω⊆^d be open and bounded such that Y⊆Ω.
By prop:periodicextension, we deduce that (ϕ_n,|_Ω)_n
is bounded in H^1(Ω). Let ψ∈(Ω) with ψ=1
on Y. Then (ψϕ_n,)_n
is bounded in H_0^1(Ω). By thm:Rellich, we
find an Ł(Ω)-convergent subsequence. This sequence also converges
in Ł(Y). Since ψ=1 on Y, we obtain the assertion.


Next, we provide a Poincaré-type inequality for the periodic gradient. 


 There exists c≥ 0 such that for
all u∈ H_♯^1(Y)

    u-∫_Yu_Ł(Y)≤ c _♯u_Ł(Y)^d.

 In particular, (_♯)⊆Ł(Y)^d is closed, (_♯)={_Y} and the operator
 
    _♯ H^1_♯(Y)∩{_Y}^→(_♯)

is an isomorphism.



The proof is left as exer:poincare.

We are now in a position to formulate the particular example we have in mind.
Problems of this type with highly oscillatory coefficients are also
referred to as homogenisation problems. We refer to the comments section
for more details on this.
[Homogenisation problem for the wave equation]
 Let c>0, a^d→^d× d be bounded,
measurable, a(x)=a(x)^*≥ c for all x∈^d. Furthermore,
assume that a is 01^d-periodic. Let ν>0, f∈(;Ł(Y))
and for n∈ consider the problem of finding u_n∈(;Ł(Y))
such that 

    ν^2u_n-_♯a(n)_♯u_n=f.

We have already established that there exists a uniquely determined solution, u_n. 
Employing the same trick as in sec:Three_Exponentially_Stable_Models_for_Heat_Conduction,
we rewrite eq:waven using v_nνu_n, the canonical embedding ι_♯(_♯)↪Ł(Y)^d
as well as q_n-ι_♯^*a(n)ι_♯ι_♯^*_♯u_n
to obtain

    (ν[                 1                 0;                 0 (ι_♯^*a(n)ι_♯)^-1 ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])[ v_n; q_n ]=[ f; 0 ].

Note that we have used that (ι_♯^*a(n)ι_♯)(_♯)→(_♯)
is continuously invertible and strictly positive definite (uniformly
in n); see prop:exp_stab_heat. Also note that ι_♯^*a(n)ι_♯
is selfadjoint. As in  exer:closed_range we see that (ι_♯^*_♯)^*=-_♯ι_♯.
Thus, the operator

    S^(n)(ν[                 1                 0;                 0 (ι_♯^*a(n)ι_♯)^-1 ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])^-1

 is well-defined and bounded in (;Ł(Y)×(_♯)).
We aim to find the limit of (S^(n))_n as n→∞. For this,
we want to apply thm:convPDE. We readily see using thm:RellichII
and exer:compactness of adjoint that 

    A[       0   _♯ι_♯; ι_♯^*_♯       0 ]

satisfies the assumptions in thm:convPDE. Thus, it is
left to analyse ((ι_♯^*a(n)ι_♯)^-1)_n.
This is the subject of the next section. For this reason, we define

    𝔞_n:=(ι_♯^*a(n)ι_♯)^-1  (n∈).






§ THE LIMIT OF THE SCALED COEFFICIENT SEQUENCE


In this section, we shall apply our earlier findings to higher-dimensional
problems. Again, we fix Y01^d as well as ι_♯(_♯)↪Ł(Y)^d, the canonical embedding. Before we are able to present the central result of this section, we need a preliminary result.

Throughout, let a^d→^d× d
be measurable, bounded and 01^d-periodic such that a(x)≥ c for each x∈^d for some c>0.


 
Let ξ∈^d. Then there exists a unique v_ξ∈Ł(Y)^d with v_ξ-ξ∈(_♯) and a()v∈(_♯).



Take w∈ H^1_♯(Y) such that

    _♯w=-ι_♯(ι_♯^*a()ι_♯)^-1ι_♯^*a()ξ = -ι_♯𝔞_n ι_♯^*a()ξ.

This is possible, since the right-hand side belongs to (_♯) by definition. We put v_ξ_♯w+ξ. Then
v_ξ-ξ∈(_♯) and we have

    ι_♯^*a()v_ξ    =ι_♯^*a()(_♯w+ξ)
     =ι_♯^*a()(-ι_♯𝔞_nι_♯^*a()ξ+ξ)
        =-ι_♯^*a()ι_♯𝔞_nι_♯^*a()ξ+ι_♯^*a()ξ=0.

The latter gives a()v_ξ∈(_♯)^=(_♯). For the uniqueness, we assume v∈(_♯)
with a()v∈(_♯). 
Then

    (ι_♯^* a()ι_♯)ι_♯^∗ v=ι_♯^*a()v=0,

which implies ι_♯^*v=0 since ι_♯^*a()ι_♯
is invertible. Thus v=0.

The previous result induces the linear mapping 

    a_^d∋ξ↦∫_Yav_ξ∈^d,

where v_ξ∈Ł(Y)^d is the unique vector field from lem:ExistenceLemma.


We gather some elementary facts on a_. 

 
  * 
  We have (a^∗)_=a_^∗. In particular, if a is pointwise selfadjoint then so is a_. Indeed, let ξ,ζ∈^d
and v_ξ and v_ζ∈Ł(Y)^d be the corresponding functions for a^∗ and a, respectively, according to lem:ExistenceLemma. Then there exist w_ξ,w_ζ∈(_♯) with v_ξ-ξ=_♯ w_ξ and v_ζ-ζ=_♯ w_ζ. 
We compute

    (a^∗)_ξζ_^d    =∫_Y(a^∗ v_ξ)(y)v_ζ(y)-_♯ w_ζ(y)_^dỵ
        =∫_Y(a^∗ v_ξ)(y)v_ζ(y)_^dỵ-∫_Y(a^∗ v_ξ)(y)_♯ w_ζ(y)_^dỵ
        =∫_Yv_ξ(y)(av_ζ)(y)_^dỵ-a^∗ v_ξ_♯ w_ζ_Ł(Y)^d
        =∫_Yv_ξ(y)(av_ζ)(y)_^dỵ
        =∫_Y_♯ w_ξ(y)+ξ(av_ζ)(y)_^dỵ
        =∫_Yξ(av_ζ)(y)_^dỵ=ξa_ζ_^d.


  * a_ is strictly positive definite. As above, one shows

    ξa_ξ_^d=∫_Yv_ξ(y)(av_ξ)(y)_^dỵ≥ cv_ξ^2_Ł(Y)^d   (ξ∈^d)

and since the right-hand side is strictly positive if ξ 0 by lem:ExistenceLemma, we derive the assertion.



The construction of a_ now allows us to formulate
the main result of this section.


 We have

    𝔞_n = (ι_♯^*a(n)ι_♯)^-1→(ι_♯^*a_ι_♯)^-1𝔞_  (n→∞)

in the weak operator topology of ((_♯)).


The proof of thm:homo requires some more preparation.
One of the results needed is a variant of thm:periodic_homo
for Ł(Y). However, it will be beneficial to finish exa:wave_homo first.
[exa:wave_homo continued]
 The operator sequence (S^(n))_n converges in the strong operator
topology of ((;Ł(Y)×(_♯))) to the
following limit 

    (ν[  1  0;  0 𝔞_ ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])^-1.




 Let f^d→ be measurable and 01^d-periodic.
Let Ω⊆^d be open, bounded and assume f|_Y∈Ł(Y).
Then 

    f(n·)→(∫_Y f) _Ω

weakly in Ł(Ω) as n→∞.


 Due to the boundedness of Ω we find a finite
set F⊆^d such that Ω⊆⋃_k∈ Fk+Y.
Thus, by periodicity, it suffices to restrict our attention to the case when Ω=Y. 
First of all we show that (f(n·))_n is a bounded sequence
in Ł(Y). For this, we compute  

    ∫_Yf(nx)^2x̣    =1/n^d∫_nYf(y)^2ỵ
     =1/n^dn^d∫_Yf(y)^2ỵ=f_Ł(Y)^2,

where we used periodicity again. This chain of equalities also shows with the help of a density argument, that it suffices to assume f|_Y to be a simple function. Note that in this case f ∈ L_∞(^d). 

Finally, we note that by thm:periodic_homo 

    f(n·)g_Ł(Y)→(∫_Y f)_Yg_Ł(Y)   (n→∞)

for each g∈Ł(Y)⊆ L_1(Y), which implies the desired assertion.




Let
(q_n)_n and (r_n)_n be weakly convergent
sequences in a Hilbert space H with weak limits q,r∈ H, respectively. Moreover,  let X⊆ H be a closed subspace and ι X→ H the canonical embedding. Assume that

    q_n∈ X for each n∈ and (ι^*r_n)_n is strongly convergent in X.

Then

    lim_n→∞r_nq_n_H=rq_H.




Since ι^∗ H→ X is continuous it is also weakly continuous, and thus, 

    ι^∗ r_n →ι^∗ r   (n→∞)

strongly in X. For n∈ we compute

    r_nq_n_H    =r_nιι^*q_n_H=ι^*r_nι^*q_n_X
     →ι^∗ rι^*q_X.

Since X is a closed subspace, it is also weakly closed and thus q∈ X which yields

    ι^∗ rι^*q_X=rq_H.


The next theorem is a version of the so-called `div-curl lemma'.

Let (q_n)_n and (r_n)_n be weakly
convergent sequences in Ł(Y)^d to some q,r∈Ł(Y)^d, respectively.
Assume that

    q_n∈(_♯)  for each  n∈ and (ι_♯^*r_n)_n is strongly convergent in (_♯).

Then

    ∫_Yr_n(x)q_n(x)_^dϕ(x)x̣→∫_Yr(x)q(x)_^dϕ(x) x̣

for all ϕ∈(Y) as n→∞. 



Let ϕ∈(Y), n∈. Since q_n ∈(_♯), we find a unique w_n∈ H^1_♯(Y) with w_n ∈{_Y}^=(_♯)^ such that 

    _♯w_n=q_n.

Moreover, since _♯ H^1_♯(Y)∩{_Y}^→(_♯) is an isomorphism by prop:Poincare_periodic, we infer that (w_n)_n is a weakly convergent sequence in H^1_♯(Y) and denote its weak limit by w∈ H^1_♯(Y).  By thm:RellichII, we deduce w_n→ w
strongly in Ł(Y)^d. Moreover, note that (ϕ w_n)_n
weakly converges to ϕ w in H_♯^1(Y). In
particular, _♯(ϕ w_n)→_♯(ϕ w)
weakly in Ł(Y)^d.  For n∈,  we compute

    ∫_Yr_n(x)q_n(x)_^dϕ(x)x̣   = r_nq_n ϕ_L(Y)^d 
    =r_n(_♯ w_n) ϕ_L(Y)^d
       =r_n_♯ (ϕ w_n)_L(Y)^d- r_nw_n _♯ϕ_Ł(Y)^d.

Now, the first term on the right-hand side of this equality tends to r_♯ (ϕ w)_Ł(Y)^d by 
lem:dcl_proto applied to X=(_♯), which is
closed by prop:Poincare_periodic. The second term tends to rw _♯ϕ_Ł(Y)^d by strong convergence of (w_n)_n and weak
convergence of (r_n)_n in Ł(Y)^d. Thus, we obtain

    ∫_Yr_n(x)q_n(x)_^dϕ(x)x̣   →r_♯(ϕ w)_Ł(Y)^d-rw _♯ϕ_Ł(Y)^d
       =∫_Yr(x)q(x)_^dϕ(x) x̣  (n→∞).



We will apply the latter theorem to the concrete case when r_n=a(n) q_n in order to determine the weak limit of (a(n)q_n)_n.  


Let (q_n)_n and (a(n)q_n)_n
be weakly convergent in Ł(Y)^d to some q and r, respectively.
Assume that

    q_n∈(_♯)  for each  n∈ and (ι_♯^∗ a(n)q_n)_n is strongly convergent in (_♯).

Then r=a_q. 



Let ξ∈^d and choose v v_ξ∈Ł(Y)^d according to lem:ExistenceLemma for a^∗ instead of a; that is, v-ξ∈(_♯) and a^∗() v ∈ (_♯).  For n∈, we define
v_n v_(n·)∈Ł(Y)^d.  Next, let g∈ C_♯^∞(Y). Then
we compute

    a^∗(n)v_n_♯g_Ł(Y)^d    =∫_Ya^∗(nx)v_(nx)_♯g(x)_^dx̣
        =1/n^d∫_nYa^∗(y)v_(y)(_♯g)(y/n)_^dỵ
       = 1/n^d-1∫_nYa^∗(y)v_ (y)( g(·/n))(y)_^dỵ.

In order to compute the last integral, we employ lem:C_sharp_core and rem:approx_in_Kern_div_sharp to find a sequence (ϕ_k)_k∈ in C_♯^∞(Y)^d∩(_♯) such that ϕ_k → a^∗()v as k→∞ in Ł(Y)^d. The latter implies
(ϕ_k)_→ a^∗() v_ as k→∞ in Ł(nY)^d for each n∈ and (ϕ_k)_ = 0 for all k∈ by prop:periodicextension.  
Thus, we obtain with integration by parts (note that the boundary terms vanish due to the periodicity of ϕ_k and g)

    a^∗(n)v_n_♯g_Ł(Y)^d   =1/n^d-1a^∗()v_( g(·/n))_Ł(nY)^d
       =1/n^d-1lim_k→∞(ϕ_k)_( g(·/n))_Ł(nY)^d=0.

Since C_♯^∞(Y) is a core for _♯, we infer that a^∗(n)v_n ∈(_♯)^ and hence,

    ι_♯^∗ a^∗(n)v_n=0   (n∈).

Moreover, we have a^∗(n)v_n →∫_Y a^∗ v=(a^∗)_ξ weakly in Ł(Y) as n→∞ by lem:homo. Thus, by thm:dcl_abst applied to q_n and r_n a^∗(n)v_n, we deduce that for all ϕ∈(Y)

    lim_n→∞∫_Y a^∗(nx)v_n(x)q_n(x)_^dϕ(x)x̣= ∫_Y (a^∗)_ξq(x)_^dϕ(x) x̣.

On the other hand, v_n →(∫_Y v)_Y =ξ_Y weakly in Ł(Y) as n→∞ by lem:homo, where ∫_Yv=ξ follows from v-ξ∈(_♯). Thus, we can apply thm:dcl_abst to q_n v_n and r_n a(n)q_n and obtain for all ϕ∈(Y)

    lim_n→∞∫_Y a^∗(nx)v_n(x)q_n(x)_^dϕ(x)x̣   =lim_n→∞∫_Y v_n(x)a(nx) q_n(x)_^dϕ(x)x̣
       = ∫_Y ξr(x)_^dϕ(x) x̣.

Thus, we have

    ∫_Y (a^∗)_ξq(x)_^dϕ(x) x̣=∫_Y ξr(x)_^dϕ(x) x̣

for each ϕ∈(Y). Hence, we infer

    ξr(x)_^d=(a^∗)_ξq(x)_^d=ξa_  q(x)_^d

for almost every x∈ Y, where we have used rem:a_hom<ref>. Since the latter holds for each ξ∈^d, we deduce r=a_ q.



Let n∈ and for u∈(_♯) we put q_n𝔞_n u.
We need to show that (q_n)_n weakly converges to 𝔞_ u.
For this, we choose subsequences (without relabeling) such that both
(q_n)_n and (a(n)q_n)_n weakly converge
 to some q and r, respectively. By definition, we have
q_n∈(_♯) and ι_♯^*a(n)q_n=u for each n∈.
Hence, by lem:JikovLemma, we deduce a_q=r. As (_♯) is closed, it is also weakly closed, and hence, q∈(_♯). Thus, we have 

    ι_♯^∗ a_ι_♯ q= ι_♯^∗ r,

or equivalently

    q= 𝔞_ι_♯^∗ r.

Now, since u=ι_♯^*a(n)q_n→ι_♯^∗ r weakly, we infer

    q=𝔞_ u.

A subsequence argument now yields the claim.
































































































































§ COMMENTS


The theory of finding partial differential equations as appropriate
limit problems of partial differential equations with highly oscillatory
coefficients is commonly referred to as `homogenisation'. The mathematical
theory of homogenisation goes back to the late 1960s and early 70s.
We refer to <cit.> as an early monograph wrapping
up the available theory to that date.

The usual way of addressing homogenisation problems is to look at
static (i.e., time-independent) problems first. The corresponding
elliptic equation is then intensively studied. Even though it might
be hidden in the derivations above, the `study of the elliptic problem'
essentially boils down to addressing the limit behaviour of 𝔞_n
as n→∞; see <cit.>. Consequently, generalisations
of the periodic case have been introduced. The periodic case (and
beyond) is covered in <cit.>; non-periodic
cases and corresponding notions have been introduced in <cit.>
and, independently, in <cit.>.

An important technical tool to obtain results in this direction is
the - lemma or the notion of `compensated
compactness'. In the above presented material, this is thm:dcl_abst;
the main difficulty to overcome is that of finding a limit of a product (q_nr_n)_n
of weakly convergent sequences (q_n)_n,(r_n)_n
in Ł(Ω)^3 for some open Ω⊆^3. It turns
out that if ( q_n)_n and ( r_n)_n
converge strongly in an appropriate sense, then ∫_Ωq_nr_nϕ
converges to the desired limit for all ϕ∈(Ω). In
thm:dcl_abst the -condition is strengthened in
as much as we ask q_n to be a gradient, which results in q_n=0.
The -condition is replaced by the condition involving ι_♯^*,
which can in fact be shown to be equivalent, see <cit.>. The restriction to periodic boundary
value problems is a mere convenience. It can be shown that the arguments
work similarly for non-periodic boundary conditions, and even with the same
limit, see <cit.>.

There are many generalisations of the - lemma. For
this, we refer to <cit.> (and the references given
there) and to the rather recently found operator-theoretic perspective,
with plenty of applications not solely restricted to the operators  and , see <cit.>.

The way of deriving the homogenised equation (i.e., the limit of 𝔞_n)
is akin to some derivations in <cit.>. Further
reading on homogenisation problems can also be found in these references.
The first step of combining homogenisation processes and evolutionary
equations has been made in <cit.> and has had some profound
developments for both quantitative and qualitative results; see <cit.>.



§ EXERCISES

tocsectionExercises


Under the same assumptions of thm:convPDE
show 

    ((νM_n(ν)+A)^-1 - (νM(ν)+A)^-1)ν^-1_((;H))→0.




 Let Ω⊆^d be open and ε >0. We define the set 

    Ω_εx∈Ω(x,∂Ω)>ε.


 
  *  Let (ϕ_k)_k∈ in (^d) be a δ-sequence (cf. exer:delta-seq) and u∈ H^1(Ω). We identify each function on Ω by its extension to ^d by 0. Prove that for k∈ large enough, ϕ_k∗ u∈ H^1(Ω_ε) with 
 
    (ϕ_k∗ u)=ϕ_k∗ u Ω_ε.


  *  Use <ref> to prove lem:kernel_grad.




 Prove the `subsequence argument': Let X be a topological space and (x_n)_n a sequence in X. Assume that there exists x∈ X such that each subsequence of (x_n)_n has a subsequence converging to x. Show that x_n→ x as n→∞.




Let H_0, H_1 be Hilbert spaces and C(C)⊆ H_0→ H_1 be a closed linear operator such that (C)↪ H_0 compactly. Let P_(C)^ H_0→ H_0 denote the orthogonal projection onto the closed subspace (C)^. Prove that there exists c>0 such that 

    ∀ u∈(C): P_(C)^ u_H_0≤ c Cu_H_1.

Apply this result to prove prop:Poincare_periodic.



Let H_0,H_1 be Hilbert
spaces. Let C(C)⊆ H_0→ H_1 be closed and
densely defined. Assume that (C)∩(C)^↪ H_0
compactly. Show that, then, (C^*)∩(C^*)^↪ H_1
compactly.



Let ν>0, Ω=01^d, s∈ L_∞() be 1-periodic, 0≤ s≤ 1,
and a as in exa:wave_homo. Show that
(u_n)_n in (;Ł(Y)) satisfying 

    ν^2s(n)u_n+ν(1-s(n))u_n-_♯a(n)_♯u_n=f

for some f∈(;Ł(Y)) is convergent to some u∈(;Ł(Y)).
Which limit equation is satisfied by u?



Let (α_n)_n be a null-sequence in 01 and let
a be as in exa:wave_homo. Show 

    ([        ν        0;        0 ν^α_n𝔞_n ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])^-1
       →([  ν  0;  0 𝔞_ ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])^-1

in the strong operator topology. Show that if f∈-μ(;Ł(Y)_),
where Ł(Y)_ϕ∈Ł(Y)∫_Yϕ=0 for
some small enough μ>0, we have

    ([  ν  0;  0 𝔞_ ]+[       0   _♯ι_♯; ι_♯^*_♯       0 ])^-1[ f; 0 ]∈-μ(;Ł(Y)×(_♯)).




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