The Caccioppoli Ultrafunctions
Vieri Benci Luigi Carlo Berselli Carlo Romano Grisanti
math.AP 35D05, 46F30, 03H05, 26E30






The Caccioppoli Ultrafunctions 
    Vieri Benci, Luigi Carlo Bersellicorresponding author,
and Carlo Romano Grisanti
    
====================================================================================

Dipartimento di Matematica, Università degli Studi di Pisa, Via F.
Buonarroti 1/c, 56127 Pisa, ITALY 

Ultrafunctions are a particular class of functions defined on a hyperreal
field ℝ^∗⊃ℝ. They have been introduced
and studied in some previous works (<cit.>,<cit.>,<cit.>).
In this paper we introduce a particular space of ultrafunctions which
has special properties, especially in term of localization of functions
together with their derivatives. An appropriate notion of integral
is then introduced which allows to extend in a consistent way the
integration by parts formula, the Gauss theorem and the notion of
perimeter. This new space we introduce, seems suitable for applications
to PDE's and Calculus of Variations. This fact will be illustrated
by a simple, but meaningful example.

Keywords. Ultrafunctions,
Non Archimedean Mathematics, Non Standard Analysis, Delta function,
distributions. 





§ INTRODUCTION


The Caccioppoli ultrafunctions can be considered as a kind generalized
functions. In many circumstances, the notion of real function is not
sufficient to the needs of a theory and it is necessary to extend
it. Among people working in partial differential equations, the theory
of distributions of L. Schwartz is the most commonly used, but other
notions of generalized functions have been introduced by J.F. Colombeau
<cit.> and M. Sato <cit.>. This paper deals with
a new kind of generalized functions, called “ultrafunctions”, which
have been introduced recently in <cit.> and developed in <cit.>.
They provide generalized solutions to certain equations which do not
have any solution, not even among the distributions. 

Actually, the ultrafunctions are pointwise defined on a subset of
(ℝ^∗)^N, where ℝ^∗
is the field of hyperreal numbes, namely the numerical field on which
nonstandard analysis (NSA in the sequel) is based. We refer to Keisler
<cit.> for a very clear exposition of NSA and in the following,
starred quantities are the natural extensions of the corresponding
classical quantities.

The main novelty of this paper is that we introduce the space of Caccioppoli
ultrafunctions V_Λ(Ω). They satisfy special properties
which are very powerful in applications to Partial Differential Equations
and Calculus of Variations. The construction of this space is rather
technical, but contains some relevant improvements with respect to
the previous notions present in the literature (see e.g. <cit.>). 

The main peculiarities of the ultrafunctions in V_Λ(Ω)
are the following: there exist a generalized partial derivative D_i
and a generalized integral  (called poinwise integral) such
that 


  * the generalized derivative is a local operator namely, if supp(u)⊂ E^*
(where E is an open set), then supp(D_iu)⊂ E^*.

  * ∀ u,v∈ V_Λ(Ω), 

    D_iuv dx=- uD_iv dx  ;


  * the “generalized” Gauss theorem holds for any measurable set A
(see Theorem <ref>)

    _AD·ϕ dx=_∂ Aϕ·𝐧_A dS  ;


  * to any distribution T∈𝒟^'(Ω)
we can associate an equivalence class of ultrafunctions [u]
such that, ∀ v∈[u], ∀φ∈𝒟(Ω),

    st( vφ° dx)=⟨ T,φ⟩ ,

where st(·) denotes the standard part of an hyperreal number. 

The most relevant point, which is not present in the previous approaches
to ultrafunctions, is that we are able the extend the notion of partial
derivative so that it is a local operator and it satisfies the usual
formula valid when integrating by parts, at the price of a suitable
extension of the integral as well. In the proof of this fact, the
Caccioppoli sets play a fundamental role.

It is interesting to compare the result about the Caccioppoli ultrafunctions
with the well-known Schwartz impossibility theorem: “there
does not exist a differential algebra (𝔄,+,⊗,D)
in which the distributions can be embedded, where D is a linear
operator that extends the distributional derivative and satisfies
the Leibniz rule (namely D(u⊗ v)=Du⊗ v+u⊗ Dv)
and ⊗ is an extension of the pointwise product on 𝒞(ℝ).” 

The ultrafunctions extend the space of distributions; they do not
violate the Schwartz theorem since the Leibniz rule, in general, does
not hold (see Remark <ref>). Nevertheless, we can prove
the integration by parts rule (<ref>) and the Gauss' divergence
theorem (with the appropriate extension  of the usual integral),
which are the main tools used in the applications. These results are
a development of the theory previously introduced in <cit.>
and <cit.>.

The theory of ultrafunctions makes deep use of the techniques of NSA
presented via the notion of Λ-limit. This presentation has
the advantage that a reader, which does not know NSA, is able to follow
most of the arguments. 

In the last section we present some very simple examples to show that
the ultrafunctions can be used to perform a precise mathematical analysis
of problems which are not tractable via the distributions.



 §.§ Plan of the paper


In section <ref>, we present a summary of the theory of Λ-limits
and their role in the development of the ultrafunctions using nonstandard
methods, especially in the context of transferring as much as possible
the language of classical analysis. In Section <ref>,
we define the notion of ultrafunctions, with emphasis on the pointwise integral.
In Section <ref>, we define the most relevant
notion, namely the generalized derivative, and its connections with
the pointwise integral, together with comparison with the classical
and distributional derivative. In Section <ref>, we show
how to construct a space satisfying all the properties of the generalized
derivative and integrals. This Section is the most technical and can
be skipped in a first reading. Finally, in Section <ref>,
we present a general result and two very simple variational problem.
In particular, the second problem is very elementary but without solutions
in the standard H^1-setting. Nevertheless it has a natural and
explicit candidate as solution. We show how this can be described
by means of the language of ultrafunctions. 



 §.§ Notations


If X is a set and Ω is a subset of ℝ^N,
then 


  * 𝒫(X) denotes the power set of X and 𝒫_fin(X)
denotes the family of finite subsets of X; 

  * 𝔉(X,Y) denotes the set of all functions from
X to Y and 𝔉(Ω)=𝔉(Ω,ℝ); 

  * 𝒞(Ω) denotes the set of continuous
functions defined on Ω; 

  * 𝒞^k(Ω) denotes the set of functions
defined on Ω which have continuous derivatives up to the order
k;

  * H^k,p(Ω) denotes the usual Sobolev space of functions
defined on Ω; 

  * if W(Ω) is any function space, then W_c(Ω)
will denote de function space of functions in W(Ω)
having compact support; 

  * 𝒞_0(Ω∪Ξ),  Ξ⊆∂Ω,
denotes the set of continuous functions in 𝒞(Ω∪Ξ)
which vanish for x∈Ξ;

  * 𝒟(Ω) denotes the set of the infinitely
differentiable functions with compact support defined on Ω; 𝒟^'(Ω)
denotes the topological dual of 𝒟(Ω),
namely the set of distributions on Ω; 

  * for any ξ∈(ℝ^N)^∗,ρ∈ℝ^∗,
we set 𝔅_ρ(ξ)={ x∈(ℝ^N)^∗: |x-ξ|<ρ}; 

  * 𝔰𝔲𝔭𝔭(f)=(supp(f))^* where supp is the usual notion
of support of a function or a distribution;

  * 𝔪𝔬𝔫(x)={y∈(ℝ^N)^∗:x∼ y}
where x∼ y means that x-y is infinitesimal;

  * 𝔤𝔞𝔩(x)={y∈(ℝ^N)^∗:x-y is finite}
;

  * if W is a generic function space, its topological dual will be
denoted by W^' and the pairing by ⟨·,·⟩ _W

  * we denote by χ_X the indicator (or characteristic) function
of X, namely

    χ_X(x)=
    1     if   x∈ X
    
    0     if   x∉ X ;


  * |X| will denote the cardinality of X.




§ Λ-THEORY


In this section we present the basic notions of Non Archimedean Mathematics
and of Nonstandard Analysis, following a method inspired by <cit.>
(see also <cit.> and <cit.>).



 §.§ Non Archimedean Fields


Here, we recall the basic definitions and facts regarding non-Archimedean
fields. In the following, 𝕂 will denote an ordered field.
We recall that such a field contains (a copy of) the rational numbers.
Its elements will be called numbers. 

Let 𝕂 be an ordered field and ξ∈𝕂. We
say that: 



  * ξ is infinitesimal if, for all positive n∈ℕ, |ξ|<1/n; 

  * ξ is finite if there exists n∈ℕ such as |ξ|<n; 

  * ξ is infinite if, for all n∈ℕ, |ξ|>n (equivalently,
if ξ is not finite). 


An ordered field 𝕂 is called Non-Archimedean if it contains
an infinitesimal ξ≠0. 

It is easily seen that all infinitesimal are finite, that the inverse
of an infinite number is a nonzero infinitesimal number, and that
the inverse of a nonzero infinitesimal number is infinite. 

A superreal field is an ordered field 𝕂 that properly
extends ℝ. 

It is easy to show, due to the completeness of ℝ, that
there are nonzero infinitesimal numbers and infinite numbers in any
superreal field. Infinitesimal numbers can be used to formalize a
new notion of closeness: 

 We say that two numbers ξ,ζ∈𝕂
are infinitely close if ξ-ζ is infinitesimal. In this case,
we write ξ∼ζ. 

Clearly, the relation ∼ of infinite closeness is an equivalence
relation and we have the following theorem

If 𝕂 is a superreal field, every finite number ξ∈𝕂
is infinitely close to a unique real number r∼ξ, called the
standard part of ξ. 

Given a finite number ξ, we denote its standard part by st(ξ),
and we put st(ξ)=±∞ if ξ∈𝕂 is a positive
(negative) infinite number. 

Let 𝕂 be a superreal field, and ξ∈𝕂 a
number. The monad of ξ is the set of all numbers that are infinitely
close to it: 

    𝔪𝔬𝔫(ξ)={ζ∈𝕂:ξ∼ζ},

and the galaxy of ξ is the set of all numbers that are finitely
close to it: 

    𝔤𝔞𝔩(ξ)={ζ∈𝕂:ξ-ζ is finite}


By definition, it follows that the set of infinitesimal numbers is
𝔪𝔬𝔫(0) and that the set of finite numbers is 𝔤𝔞𝔩(0).



 §.§ The Λ-limit


In this section we introduce a particular non-Archimedean field by
means of Λ-theory [Readers expert in nonstandard analysis will recognize that Λ-theory
is equivalent to the superstructure constructions of Keisler (see
<cit.> for a presentation of the original constructions
of Keisler). ] (for complete proofs and further informations the reader is referred
to <cit.>, <cit.> and <cit.>). To recall
the basics of Λ-theory we have to recall the notion of superstructure
on a set (see also <cit.>): 

Let E be an infinite set. The superstructure on E is the set

    V_∞(E)=⋃_n∈ℕV_n(E),

where the sets V_n(E) are defined by induction setting 

    V_0(E)=E

and, for every n∈ℕ, 

    V_n+1(E)=V_n(E)∪𝒫(V_n(E)).


Here 𝒫(E) denotes the power set of E. Identifying
the couples with the Kuratowski pairs and the functions and the relations
with their graphs, it follows that V_∞(E) contains almost
every usual mathematical object that can be constructed starting with
E; in particular, V_∞(ℝ), which is the superstructure
that we will consider in the following, contains almost every usual
mathematical object of analysis.

Throughout this paper we let 

    𝔏=𝒫_fin(V_∞(ℝ))

and we order 𝔏 via inclusion. Notice that (𝔏,⊆)
is a directed set. We add to 𝔏 a point at infinity
Λ∉𝔏, and we define the following family of
neighborhoods of Λ: 

    {{Λ}∪ Q| Q∈𝒰},

where 𝒰 is a fine ultrafilter on 𝔏,
namely a filter such that 


  * for every A,B⊆𝔏, if A∪ B=𝔏
then A∈𝒰 or B∈𝒰; 

  * for every λ∈𝔏 the set Q(λ):={μ∈𝔏|λ⊆μ}∈𝒰. 

In particular, we will refer to the elements of 𝒰 as
qualified sets and we will write Λ=Λ(𝒰) when
we want to highlight the choice of the ultrafilter. A function φ: 𝔏→ E
will be called net (with values in E). If φ(λ)
is a real net, we have that


    lim_λ→Λφ(λ)=L

if and only if

    ∀ε>0, ∃ Q𝒰,  such  that  ∀λQ, |φ(λ)-L|<ε.




As usual, if a property P(λ) is satisfied by any λ
in a neighborhood of Λ, we will say that it is eventually
satisfied.

Notice that the Λ-topology satisfies these interesting properties:

If the net φ(λ) takes values in a compact
set K, then it is a converging net. 


Suppose that the net φ(λ) has a subnet converging to
L∈ℝ. We fix ε>0 arbitrarily and we have
to prove that Q_ε∈𝒰 where

    Q_ε={λ∈𝔏 | |φ(λ)-L| <ε} .

We argue indirectly and we assume that 

    Q_ε∉𝒰

 Then, by the definition of ultrafilter, N=𝔏\ Q_ε∈𝒰
and hence

    ∀λ∈ N, |φ(λ)-L|≥ε.

This contradicts the fact that φ(λ) has a subnet which
converges to L.


Assume that φ: 𝔏→ E,
where E is a first countable topological space; then if


    lim_λ→Λφ(λ)=x_0,

there exists a sequence {λ_n} in
𝔏 such that

    lim_n→∞φ(λ_n)=x_0

We refer to the sequence φ_n:=φ(λ_n) as a
subnet of φ(λ).

It follows easily from the definitions.


Let φ: 𝔏→ V be a net with value in
bounded set of a reflexive Banach space equipped with the weak topology;
then 

    v:=lim_λ→Λφ(λ),

is uniquely defined and there exists a sequence n↦φ(λ_n)
which converges to v. 




The set of the hyperreal numbers ℝ^∗⊃ℝ
is a set equipped with a topology τ such that


  * every net φ: 𝔏→ℝ has a unique
limit in ℝ^∗, if 𝔏 and ℝ^∗
are equipped with the Λ and the τ topology respectively;

  * ℝ^∗ is the closure of ℝ with respect
to the topology τ;

  * τ is the coarsest topology which satisfies the first property. 




The existence of such ℝ^∗ is
a well known fact in NSA. The limit ξ∈ℝ^∗
of a net φ: 𝔏→ℝ
with respect to the τ topology, following <cit.>,
is called the Λ-limit of φ
and the following notation will be used:

    ξ=lim_λ↑Λφ(λ)

namely, we shall use the up-arrow “↑” to remind that
the target space is equipped with the topology τ.

Given 

    ξ:=lim_λ↑Λφ(λ)  and  η:=lim_λ↑Λψ(λ),

we set

    ξ+η:=lim_λ↑Λ(φ(λ)+ψ(λ)),

and 

    ξ·η:=lim_λ↑Λ(φ(λ)·ψ(λ)).


Then the following well known theorem holds:

The definitions (<ref>) and (<ref>) are well posed and
ℝ^*, equipped with these operations, is a non-Archimedean
field.


We observe that the field of hyperreal numbers is defined as a sort
of completion of the real numbers. In fact ℝ^* is isomorphic
to the ultrapower

    ℝ^𝔏/ℑ

where

    ℑ={φ:𝔏→ℝ | φ(λ)=0   eventually}

The isomorphism resembles the classical one between the real numbers
and the equivalence classes of Cauchy sequences. This method is well
known for the construction of real numbers starting from rationals. 




 §.§ Natural extension of sets and functions


For our purposes it is very important that the notion of Λ-limit
can be extended to sets and functions (but also to differential and
integral operators) in order to have a much wider set of objects to
deal with, to enlarge the notion of variational problem and of variational
solution. 

So we will define the Λ-limit of any bounded net of mathematical
objects in V_∞(ℝ) (a net φ: 𝔏→ V_∞(ℝ)
is called bounded if there exists n∈ℕ such that, ∀λ∈𝔏,φ(λ)∈ V_n(ℝ)).
To do this, let us consider a net

    φ:𝔏→ V_n(ℝ).

We will define lim_λ↑Λφ(λ) by
induction on n. 

For n=0, lim_λ↑Λφ(λ) is defined
by (<ref>). By induction we may assume that the limit
is defined for n-1 and we define it for the net (<ref>) as
follows:

    lim_λ↑φ(λ)={lim_λ↑ψ(λ) | ψ:𝔏→ V_n-1(ℝ), ∀λ∈𝔏, ψ(λ)∈φ(λ)} .


A mathematical entity (number, set, function or relation) which is
the Λ-limit of a net is called internal. 




If ∀λ∈𝔏, E_λ=E∈ V_∞(ℝ),
we set lim_λ↑Λ E_λ=E^∗,namely

    E^∗:={lim_λ↑Λψ(λ) | ψ(λ)∈ E} .

E^∗ is called the natural extension of E. 

Notice that, while the Λ-limit of a constant sequence of
numbers gives this number itself, a constant sequence of sets gives
a larger set, namely E^∗. In general, the inclusion E⊆ E^∗
is proper.

Given any set E, we can associate to it two sets: its natural extension
E^∗ and the set E^σ, where

    E^σ={ X^∗ | X∈ E} .


Clearly E^σ is a copy of E, however it might be different
as set since, in general, X^∗≠ X. 

If φ: 𝔏→ X is a net with values in
a topological space we have the usual limit 

    lim_λ→Λφ(λ),

which, by Proposition <ref>, always exists in the Alexandrov
compactification X∪{∞}. Moreover we have
that the Λ-limit always exists and it is an element of X^*.
In addition, the Λ-limit of a net is in X^σ if
and only if φ is eventually constant. If X=ℝ
and both limits exist, then 


    lim_λ→Λφ(λ)=st(lim_λ↑Λφ(λ)).


The above equation suggests the following definition.

If X is a topological space equipped with a Hausdorff
topology, and ξ∈ X^* we set

    St_X(ξ)=lim_λ→Λφ(λ),

if there is a net φ: 𝔏→ X converging
in the topology of X and such that

    ξ=lim_λ↑Λφ(λ),

and

    St_X(ξ)=∞

otherwise.



By the above definition we have that 

    lim_λ→Λφ(λ)=St_X(lim_λ↑Λφ(λ)).



Let 

    f_λ: E_λ→ℝ,  λ∈𝔏,

be a net of functions. We define a function

    f:(lim_λ↑Λ E_λ)→ℝ^∗

as follows: for every ξ∈(lim_λ↑Λ E_λ)
we set

    f(ξ):=lim_λ↑Λ f_λ(ψ(λ)),

where ψ(λ) is a net of numbers such that 

    ψ(λ)∈ E_λ  and  lim_λ↑Λψ(λ)=ξ.

A function which is a Λ-limit is called internal.
In particular if, ∀λ∈𝔏, 

    f_λ=f,    f: E→ℝ,

we set 

    f^∗=lim_λ↑Λ f_λ.

f^∗:E^∗→ℝ^∗ is called the natural
extension of f. If we identify f with its graph, then f^∗
is the graph of its natural extension.




 §.§ Hyperfinite sets and hyperfinite sums


An internal set is called hyperfinite if it is the Λ-limit
of a net φ:𝔏→𝔉 where 𝔉
is a family of finite sets. 

For example, if E∈ V_∞(ℝ), the set

    E=lim_λ↑Λ(λ∩ E)

is hyperfinite. Notice that

    E^σ⊂E⊂ E^*

so, we can say that every set is contained in a hyperfinite set. 



It is possible to add the elements of an hyperfinite set of numbers
(or vectors) as follows: let

    A:= lim_λ↑ΛA_λ,

be an hyperfinite set of numbers (or vectors); then the hyperfinite
sum of the elements of A is defined in the following way: 

    ∑_a∈ Aa= lim_λ↑Λ∑_a∈ A_λa.

In particular, if A_λ={ a_1(λ),...,a_β(λ)(λ)}with β(λ)∈ℕ,then
setting 

    β= lim_λ↑Λ β(λ)∈ℕ^∗,

we use the notation

    ∑_j=1^βa_j= lim_λ↑Λ∑_j=1^β(λ)a_j(λ).





§ ULTRAFUNCTIONS




 §.§ Caccioppoli spaces of ultrafunctions


Let Ω be an open bounded set in ℝ^N, and let
W(Ω) be a (real or complex) vector space such that 𝒟(Ω)⊆ W(Ω)⊆ L^1(Ω).

 A space of ultrafunctions modeled over the space
W(Ω) is given by

    W_Λ(Ω):=lim_λ↑ΛW_λ(Ω)={lim_λ↑Λf_λ  |  f_λ∈ W_λ(Ω)} ,

where W_λ(Ω)⊂ W(Ω) is an increasing net
of finite dimensional spaces such that

    W_λ(Ω)⊇ Span(W(Ω)∩λ).




So, given any vector space of functions W(Ω), the space of
ultrafunction generated by { W_λ(Ω)}
is a vector space of hyperfinite dimension that includes W(Ω)^σ,
as well as other functions in W(Ω)^*. Hence the ultrafunctions
are particular internal functions

    u: Ω^*→ℝ^∗.



Given a space of ultrafunctions W_Λ(Ω), a σ-basis
is an internal set of ultrafunctions {σ_a(x)} _a∈Γ
such that, Ω⊂Γ⊂Ω^* and ∀ u∈ W_Λ(Ω),
we can write

    u(x)=∑_a∈Γu(a)σ_a(x).

It is possible to prove (see e.g. <cit.>) that every space
of ultrafunctions has a σ-basis. Clearly, if a,b∈Γ
then σ_a(b)=δ_ab where δ_ab denotes the
Kronecker delta. 

Now we will introduce a class of spaces of ultrafunctions suitable
for most applications. To do this, we need to recall the notion of
Caccioppoli set:

A Caccioppoli set E is a Borel set such that χ_E∈ BV,
namely such that ∇(χ_E) (the distributional gradient
of the characteristic function of E) is a finite Radon measure
concentrated on ∂ E. 

The number

    p(E):=⟨ |∇(χ_E)|, 1⟩

is called Caccioppoli perimeter of E. From now on, with some abuse
of notation, the above expression will be written as follows:

    ∫|∇(χ_E)|dx;

this expression makes sense since “|∇(χ_E)|dx” is
a measure.

If E⊂Ω is a measurable set, we define the
density function of E as follows:

    θ_E(x)=st(m(B_η(x)∩ E^*)/m(B_η(x)∩(Ω)^*)),

where η is a fixed infinitesimal and m is the Lebesgue measure.

Clearly θ_E(x) is a function whose value is 1 in int(E)
and 0 in ℝ^N∖E; moreover, it is easy
to prove that θ_E(x) is a measurable function and we have
that

    ∫θ_E(x)dx=m(E);

also, if E is a bounded Caccioppoli set,

    ∫|∇θ_E|dx=p(E).



A set E is called special Caccioppoli set
if it is open, bounded and m(∂ E)=0. The family
of special Caccioppoli sets will be denoted by ℭ(Ω).

Now we can define a space V(Ω) suitable for our aims: 

A function f∈ V(Ω) if and only if

    f(x)=∑_k=1^nf_k(x)θ_E_k(x)

where f_k∈𝒞(ℝ^N), E_k∈ℭ(),
and n is a number which depends on f. Such a function, will
be called Caccioppoli function.

Notice that V(Ω) is a module over the ring 𝒞(Ω)
and that, ∀ f∈ V(Ω),

    (∫|f(x)| dx=0)⇒(∀ x∈ℝ^N, f(x)=0).

Hence, in particular, 

    (∫|f(x)|^2dx=0)^1/2,

is a norm (and not a seminorm).

 V_Λ(Ω) is called Caccioppoli
space of ultrafunctions if it satisfies the following properties:



  * V_Λ(Ω) is modeled on the space V(Ω) ;

  * V_Λ(Ω) has a σ-basis {σ_a(x)} _a∈Γ,
Γ⊂(ℝ^N)^*, such that ∀ a∈Γ
the support of σ_a is contained in 𝔪𝔬𝔫(a).


Notice 

The existence of a Caccioppoli space of ultrafunctions will be proved
in Section <ref>.

Usually in the study of PDE's, the function space where to work depends
on the problem or equation which we want to study. The same fact is
true in the world of ultrafunctions. However, the Caccioppoli space
V_Λ(Ω) have a special position since it satisfies
the properties required by a large class of problems. First of all
V_Λ(Ω)⊂(L^1(Ω))^*. This
fact allows to define the pointwise integral (see next sub-section)
for all the ultrafunctions. This integral turns out to be a very good
tool. However, the space L^1 is not a good space for modeling
ultrafunctions, since they are defined pointwise while the functions
in L^1 are defined a.e. Thus, we are lead to the space L^1(Ω)∩𝒞(Ω),
but this space does not contain functions such as f(x)θ_E(x)
which are important in many situations; for example the Gauss' divergence
theorem can be formulated as follows

    ∫∇· F(x)θ_E(x)dx=∫_∂ E𝐧· F(x)dS  ,

whenever the vector field F and E are sufficiently smooth. Thus
the space V_Λ(Ω) seems to be the right space for a
large class of problems.




 §.§ The pointwise integral


From now on we will denote by V_Λ(Ω) a fixed Caccioppoli
space of ultrafunctions and by {σ_a(x)} _a∈Γ
a fixed σ-basis as in Definition <ref>. If u∈ V_Λ(Ω),
we have that

    ∫^*u(x)dx=∑_a∈Γu(a)η_a  ,

where

    η_a:=∫^*σ_a(x) dx.


The equality (<ref>) suggests the following definition:

For any internal function g : Ω^*→ℝ^*,
we set 

    g(x)dx:=∑_q∈Γg(q)η_q .

In the sequel we will refer to  as to the pointwise
integral. 

From Definition <ref>, we have that 

    ∀ u∈ V_Λ(Ω), ∫^*u(x)dx= u(x)dx,

and in particular,


    ∀ f∈ V(Ω), ∫ f(x)dx= f^*(x)dx.

But in general these equalities are not true for L^1 functions.
For example if

    f(x)=
    1     if x=x_0∈Ω,
    
    0     if x≠ x_0 ,

we have that ∫^*f^*(x)dx=∫ f(x)dx=0, while f^*(x)dx=η_x_0>0.
However, for any set E∈ℭ(Ω) and any
function f∈𝒞(Ω)

    f^*(x)θ_E(x) dx=∫_Ef(x) dx,

in fact

    f^*(x)θ_E(x) dx=∫^*f^*(x)θ_E(x) dx=∫ f(x)θ_E(x) dx=∫_Ef(x) dx.

Then, if f(x)≥0 and E is a bounded open set, we have that

    f^*(x)χ_Edx< f^*(x)θ_E(x) dx< f^*(x)χ_Edx.

since

    χ_E<θ_E<χ_E.


As we will see in the following part of this paper, in many cases,
it is more convenient to work with the pointwise integral 
rather than with the natural extension of the Lebesgue integral ∫^*. 

If ∂ E is smooth, we have that ∀ x∈∂ E, θ_E(x)=1/2
and hence, if E is open,

    f^*(x)χ_E(x) dx     = f^*(x)θ_E(x) dx-1/2 f^*(x)χ_∂ E(x) dx
        =∫_Ef(x) dx-1/2 f^*(x)χ_∂ E(x) dx,

and similarly

    f^*(x)χ_E(x) dx=∫_Ef(x) dx+1/2 f^*(x)χ_∂ E(x) dx;

of course, the term 1/2 f^*(x)χ_∂ E(x) dx
is an infinitesimal number and it is relevant only in some particular
problems.



The pointwise integral allows us to define the following scalar product:

    u(x)v(x)dx=∑_q∈Γu(q)v(q)η_q.


From now on, the norm of an ultrafunction will be given by 

    ‖ u‖ =(|u(x)|^2 dx)^1/2.

Notice that 

    u(x)v(x)dx=∫^*u(x)v(x)dx⇔ uv∈ V_Λ(Ω).



If {σ_a(x)} _a∈Γ
is a σ, then

    {σ_a(x)/√(η_a)} _a∈Γ

is a orthonormal basis with respect to the scalar product (<ref>).
Hence for every u∈ V_Λ(Ω),

    u(x)=∑_q∈Γ1/η_q( u(ξ)σ_q(ξ)dξ)σ_q(x).

Moreover, we have that


By (<ref>), we have that

    σ_a(x)σ_b(x)dx=∑_q∈Γσ_a(q)σ_b(q)η_q=∑_q∈Γδ_aqδ_bqη_q=δ_abη_a,

then the result. By the above equality, taking b=a we get (<ref>). 




 §.§ The δ-bases


Next, we will define the delta ultrafunctions:

Given a point q∈Ω^∗, we denote by δ_q(x)
an ultrafunction in V_Λ(Ω) such that 

    ∀ v∈ V_Λ(Ω),  v(x)δ_q(x) dx=v(q),

and δ_q(x) is called delta (or the Dirac) ultrafunction
concentrated in q. 

Let us see the main properties of the delta ultrafunctions:

 The delta ultrafunction satisfies the following properties:



  * For every q∈Ω^∗ there exists an unique delta
ultrafunction concentrated in q;

  * for every a, b∈Ω^*, δ_a(b)=δ_b(a);

  * ‖δ_q‖ ^2=δ_q(q). 



1. Let { e_j} _j=1^β be an orthonormal real
basis of V_Λ(Ω), and set 

    δ_q(x)=∑_j=1^βe_j(q)e_j(x).

Let us prove that δ_q(x) actually satisfies (<ref>).
Let v(x)=∑_j=1^βv_je_j(x) be any ultrafunction.
Then

    v(x)δ_q(x)dx     =    (∑_j=1^βv_je_j(x))(∑_k=1^βe_k(q)e_k(x))dx=
        =    ∑_j=1^β∑_k=1^βv_je_k(q) e_j(x)e_k(x)dx=
        =    ∑_j=1^β∑_k=1^βv_je_k(q)δ_jk=∑_j=1^βv_ke_k(q)=v(q).

So δ_q(x) is a delta ultrafunction concentrated in q.
It is unique: infact, if γ_q(x) is another delta ultrafunction
concentrated in q, then for every y∈Ω^*
we have:

    δ_q(y)-γ_q(y)=(δ_q(x)-γ_q(x))δ_y(x)dx=δ_y(q)-δ_y(q)=0,

and hence δ_q(y)=γ_q(y) for every y∈Ω^∗.

2.δ_a(b)=δ_a(x)δ_b(x) dx=δ_b(a).

3. ‖δ_q‖ ^2=δ_q(x)δ_q(x) dx=δ_q(q).


By the definition of Γ, ∀ a,b∈Γ, we have that

    δ_a(x)σ_b(x)dx=σ_a(b)=δ_ab.

From this it follows readily the following result 

The set {δ_a(x)} _a∈Γ (Γ⊂Ω^∗)
is the dual basis of the sigma-basis; it will be called the δ-basis
of V_Λ(Ω). 



Let us examine the main properties of the δ-basis:

The δ-basis, satisfies the following properties:



  * u(x)=∑_q∈Γ[σ_q(ξ)u(ξ)dξ]δ_q(x);

  * 

  * 



(i) is an immediate consequence of the definition of δ-basis. 

(ii) By Theorem <ref>, it follows that:

    δ_a(x)=∑_q∈Γ1/η_q(δ_a(ξ)σ_q(ξ)dξ)σ_q(x)=∑_q∈Γ1/η_qδ_aqσ_q(x)=1/η_aσ_a(x).


(iii) Is an immediate consequence of (ii).




 §.§ The canonical extension of functions


We have seen that every function f: Ω→ℝ
has a natural extension f^*: Ω^*→ℝ^*.
However, in general, f^* is not an ultrafunction; in fact, it
is not difficult to prove that the natural extension f^∗ of
a function f, is an ultrafunction if and only if f∈ V(Ω).
So it is useful to define an ultrafunction f°∈ V_Λ(Ω)
which approximates f^*. More in general, for any internal function
u: Ω^*→ℝ^*, we will define an ultrafunction
u° as follows:

If u: Ω^*→ℝ^* is an internal function,
we define u°∈ V_Λ(Ω) by the following
formula:

    u°(x)=∑_q∈Γu(q)σ_q(x);

if f: Ω→ℝ, with some abuse of notation,
we set

    f°(x)=(f^*)°(x)=∑_q∈Γf^*(q)σ_q(x).




Since Ω⊂Γ, for any internal function u, we have
that 

    ∀ x∈Ω,  u(x)=u°(x)

and

    ∀ x∈Ω^*,  u(x)=u°(x)  ⟺ u∈ V_Λ(Ω).


Notice that 

    P°: 𝔉(Ω)^*→ V_Λ(Ω)

defined by P°(u)=u° is noting
else but the orthogonal projection of u∈𝔉(Ω)^*
with respect to the semidefinite bilinear form 

    u(x)h(x)dx.



If f∈𝒞(ℝ^N),
and E∈ℭ(), then fθ_E∈ V(Ω)
and hence

    (fθ_E)^°=f^*θ_E^*




If a function f is not defined on a set S:=Ω∖Θ,
by convention, we define

    f°(x)=∑_q∈Γ∩Θ^*f^*(q)σ_q(x).





By the definition above, ∀ x∈Γ, we have that 

    (1/|x|)^°=1/|x|    if x≠0
    
    0     if x=0.


If f∈𝒞(Ω), then f°≠ f^*
unless f∈ V_Λ(Ω). Let examine what f°
looks like. 

Let f: Ω→ℝ be 
in 


Fix x_0∈ A. Since A is bounded, there exists a set E∈ℭ(Ω)
such that 

    𝔪𝔬𝔫(x_0)⊂ E^*⊂ A^*.

We have that (see Example <ref>)

    f°(x)     =∑_a∈Γf^*(a)σ_a(x)
        =∑_a∈Γf^*(a)θ_E^*(a)σ_a(x)+∑_a∈Γf^*(a)(1-θ_E^*(a))σ_a(x)
        =f^*(x)θ_E^*(x)+∑_a∈Γ∖ E^*f^*(a)(1-θ_E^*(a))σ_a(x).

Since x_0∈ E^*, θ_E^*(x_0)=1; moreover, since
𝔪𝔬𝔫(x_0)⊂ E^*, by definition <ref>,
(ii), 

    ∀ a∈Γ∖ E^*, σ_a(x_0)=σ_x_0(a)=0.

 Then

    f°(x_0)=f^*(x_0).



If f∈𝒞(Ω), then, for any x∈Ω^*, such
that |x| is finite, we get 

    f°(x)=f^*(x).





 §.§ Canonical splitting of an ultrafunction


In many applications, it is useful to split an ultrafunction u
in a part w° which is the canonical extension
of a standard function w and a part ψ which is not directly
related to any classical object.

If u∈ V_Λ(Ω), we set

    Ξ={ x∈Ω | u(x)  is infinite}

and

    w(x)=
    st(u(x))     if x∈Ω∖Ξ
    
    0     if x∈Ξ.



For every ultrafunction u consider the splitting

    u=w°+ψ

where



  * w=w_|Ω∖Ξ and w°
which is defined by Definition <ref>, is called the functional
part of u;

  * ψ:=u-w° is called the singular part
of u.

We will refer to 

    S:={ x∈Ω^* | ψ(x)0}

 as to the singular set of the ultrafunction u. 


Notice that w°, the functional part of u, may
assume infinite values, but they are determined by the values of w,
which is a standard function defined on Ω∖Ξ.

Take ε∼0, and 

    u(x)=1/x^2+ε^2.

In this case


  * w(x)=1/x^2,

  * ψ(x)=
-ε^2/x^2(x^2+ε^2)    if x≠0
1/ε^2    if x=0,

  * S:={ x∈ℝ^* | ψ(x)0}⊂𝔪𝔬𝔫(0).


We conclude this section with the following trivial propositions which,
nevertheless, are very useful in applications: 

Let W be a Banach space such that 𝒟(Ω)⊂ W⊆ L_loc^1(Ω)
and assume that 

    u=w°+ψ

 is the canonical splittingu:=lim_λ↑Λ u_λ,


    lim_n→∞u_n=w    weakly  in  W

and 

    ∀ v∈ W,  ψ v dx∼0.



    lim_n→∞‖ u_n-w‖ _W=0

then 


It is an immediate consequence of Proposition <ref>. 

If we use the notation introduced in Definition <ref>, the
above proposition can be reformulated as follows:

If u:=lim_λ↑Λ u_λ,


    w=St_W_weak(u).

If 

    w=St_W(u).


An immediate consequence of Proposition <ref> is the
following:

If w∈ L^1(Ω) then

    w°(x)dx∼ w(x)dx.



Since V_Λ(Ω) is dense in L^1(Ω) there is
a sequence u_n∈ V_Λ(Ω) which converges strongly
to w in L^1(Ω). Now set 

    u:=lim_λ↑Λ u_|λ|.

By Proposition <ref>, we have that 

    u=w°+ψ

with ‖ψ‖ _L^1*∼0. Since u and
w° are in V_Λ(Ω), then also ψ∈ V_Λ(Ω),
so that ψ dx=∫^*ψ dx∼0. Then

    u(x)dx w°(x)dx.

On the other hand, 

    u(x)dx    =∫^*u(x)dx=lim_λ↑Λ ∫ u_|λ|dx
       ∼lim_λ→Λ ∫ u_|λ|dx=lim_n→∞ ∫_Ωu_ndx= w(x)dx.





§ DIFFERENTIAL CALCULUS FOR ULTRAFUNCTIONS


In this section, we will equip the Caccioppoli space of ultrafunctions
V_Λ(Ω) with a suitable notion of derivative which
generalizes the distributional derivative. Moreover we will extend
the Gauss' divergence theorem to the environment of ultrafunctions
and finally we will show the relationship between ultrafunctions and
distributions.



 §.§ The generalized derivative


If u∈ V_Λ(Ω)∩[C^1(Ω)]^*,
then, ∂_i^*u is well defined and hence, using Definition
<ref>, we can define an operator

    D_i : V_Λ(Ω)∩[C^1(Ω)]^*→  V_Λ(Ω)

as follows

    D_iu°=(∂_i^*u)°.

However it would be useful to extend the operator D_i to all
the ultrafunctions in V_Λ(Ω) to include in the theory
of ultrafunctions also the weak derivative. Moreover such an extension
allows to compare ultrafunctions with  distributions. In this section
we will define the properties that a generalized derivative must have
(Definition <ref>) and in section <ref>, we will
show that these properties are consistent; we will do that by a construction
of the generalized derivative.

The generalized derivative 

    D_i: V_Λ(Ω)→ V_Λ(Ω)

is an operator defined on a Caccioppoli ultrafunction space V_Λ(Ω)
which satisfies the following properties:



  * V_Λ has σ-basis {σ_a(x)} _a∈Γ,
such that ∀ a∈Γ the support of D_iσ_a
is contained in 𝔪𝔬𝔫(a);

  * if u∈ V_Λ(Ω)∩[C^1(Ω)]^*,
then, 

    D_iu°=(∂_i^*u)°;


  * ∀ u,v∈ V_Λ(Ω), 

    D_iuv dx=- uD_iv dx


  * if E∈ℭ(Ω), then ∀ v∈ V_Λ(Ω),

    D_iθ_Ev dx=-∫_∂ E^*v (𝐞_i·𝐧_E) dS

where 𝐧_E is the measure theoretic unit outer normal,
integrated on the reduced boundary of E with respect to the (n-1)-Hausdorff
measure dS (see e.g. <cit.>) and (𝐞_1,....,𝐞_N)
is the canonical basis of ℝ^N.


We remark that, in the framework of the theory of Caccioppoli sets,
the classical formula corresponding to IV is the following: ∀ v∈𝒞(Ω),

    ∫∂_iθ_Ev dx=-∫_∂ Ev (𝐞_i·𝐧_E) dS.

 


The existence of a generalized derivative will be proved in section
<ref>.

Now let us define some differential operators: 


  * ∇=(∂_1,...,∂_N) will denote the usual gradient
of standard functions;

  * ∇^*=(∂_1^*,...,∂_N^*) will denote
the natural extension of the gradient (in the sense of NSA);

  * D=(D_1,...,D_N) will denote the canonical extension of the
gradient in the sense of the ultrafunctions (Definition <ref>).

Next let us consider the divergence:


  * ∇·ϕ=∂_1ϕ_1+...+∂_Nϕ_N will
denote the usual divergence of standard vector fields ϕ∈[𝒞^1(Ω)]^N;

  * ∇^*·ϕ=∂_1^*ϕ_1+...+∂_N^*ϕ_N
will denote the divergence of internal vector fields ϕ∈[𝒞^1(Ω)^*]^N;

  * D·ϕ will denote the unique ultrafunction D·ϕ∈ V_Λ(Ω)
such that, ∀ v∈ V_Λ(Ω),

    D·ϕ vdx=-ϕ(x)· Dv dx.


And finally, we can define the Laplace operator:


  * △° or D^2 will denote the Laplace
operator defined by D∘ D.




 §.§ The Gauss' divergence theorem 


By Definition <ref>, IV, for any set E∈ℭ_Λ(Ω)
and v∈ V_Λ(Ω),

    D_iθ_Ev dx=-∫_∂ E^*v (𝐞_i·𝐧_E) dS,

and by Definition <ref>, III, 

    D_iv θ_Edx=∫_∂ E^*v (𝐞_i·𝐧_E) dS.

Now, if we take a vector field ϕ=(v_1,...,v_N)∈[V_Λ(Ω)]^N,
by the above identity, we get

    D·ϕ θ_E dx=∫_∂ E^*ϕ·𝐧_E dS.

Now, if ϕ∈𝒞^1(Ω) and ∂ E
is smooth, we get the divergence Gauss' theorem:

    ∫_E∇·ϕ dx=∫_∂ Eϕ·𝐧_E dS.


Then, (<ref>) is a generalization of the Gauss' theorem
which makes sense for any set E∈ℭ_Λ(Ω).
Next, we want to generalize Gauss' theorem to any subset A⊂Ω. 

First of all we need to generalize the notion of Caccioppoli perimeter
p(E) to any arbitrary set. As we have seen in Section <ref>,
if E∈ℭ(Ω) is a special Caccioppoli
set, we have that 

    p(E)=∫|∇θ_E| dx,

and it is possible to define a (n-1)-dimensional measure dS
as follows

    ∫_∂ Ev(x) dS:=∫|∇θ_E| v(x) dx.

In particular, if the reduced boundary of E coincides with ∂ E,
we have that (see <cit.>)

    ∫_∂ Ev(x) dS=∫_∂ Ev(x) dℋ^N-1.


Then, the following definition is a natural generalization:

If A is a measurable subset of Ω, we
set

    p(A):=|Dθ_A^°| dx

and ∀ v∈ V_Λ(Ω),

    _∂ Av(x) dS:= v(x) |Dθ_A^°| dx.



Notice that

    _∂ Av(x) dS≠ v(x)χ_∂ A^°(x) dx.

In fact the left hand term has been defined as follows:

    _∂ Av(x) dS=∑_x∈Γv(x) |Dθ_A^°(x)| η_x

while the right hand term is

    v(x)χ_∂ A^°(x) dx=∑_x∈Γv(x)χ_∂ A^°(x) η_x,

in particular if ∂ A is smooth and v(x) is bounded, ∑__x∈Γv(x)χ_∂ A^°(x) η_x
is an infinitesimal number.


If A is an arbitrary measurable subset of Ω,
we have that

    D·ϕ θ_A^° dx=_∂ Aϕ·𝐧_A^°(x) dS,

where

    𝐧_A^°(x)=
    -Dθ_A^°(x)/|Dθ_A^°(x)|    if   Dθ_A^°(x)≠0
    
    0     if   Dθ_A^°(x)=0



By Theorem <ref>, III, 


    D·ϕ θ_A^°dx=-ϕ· Dθ_A^°dx,

then, using the definition of 𝐧_A^°(x)
and (<ref>), the above formula can be written as follows:

    D·ϕ θ_A^°dx=ϕ·𝐧_A^° |Dθ_A^°| dx=_∂ Aϕ·𝐧_A^° dS.


Clearly, if E∈ℭ_Λ(Ω), then

    _∂ Eϕ·𝐧_E^° dS=∫_∂ Eϕ·𝐧_E dS.



If A is the Koch snowflake, then the usual Gauss' theorem makes
no sense since p(A)=+∞; on the other hand equation (<ref>)
holds true. Moreover, the perimeter in the sense of ultrafunction
is an infinite number given by Definition <ref>. In general,
if ∂ A is a d-dimensional fractal set, it is an interesting
open problem to investigate the relation between its Hausdorff measure
and the ultrafunction “measure” dS=|Dθ_A^°| dx.




 §.§ Ultrafunctions and distributions


One of the most important properties of the ultrafunctions is that
they can be seen (in some sense that we will make precise in this
section) as generalizations of the distributions.

The space of generalized
distribution on Ω is defined as follows: 

    𝒟_G^'(Ω)=V_Λ(Ω)/N,

where 

    N={τ∈ V_Λ(Ω) | ∀φ∈𝒟(Ω), ∫τφ dx∼0} .


The equivalence class of u in V_Λ(Ω) will be denoted
by 

    [u]_𝒟.



Let [u]_𝒟 be a generalized distribution.
We say that [u]_𝒟 is a bounded generalized
distribution if, ∀φ∈𝒟(Ω), ∫ uφ^* dx  is finite.
We will denote by 𝒟_GB^'(Ω) the set of
the bounded generalized distributions.

We have the following result. 

There is a linear isomorphism 

    Φ:𝒟_GB^'(Ω)→𝒟^'(Ω),

defined by 

    ⟨Φ([u]_𝒟),φ⟩ _𝒟(Ω)=st( u φ^∗ dx).



For the proof see e.g. <cit.>.

From now on we will identify the spaces 𝒟_GB^'(Ω)
and 𝒟^'(Ω); so, we will identify [u]_𝒟
with Φ([u]_𝒟) and we will
write [u]_𝒟∈𝒟^'(Ω)
and 

    ⟨[u]_𝒟,φ⟩ _𝒟(Ω):=⟨Φ[u]_𝒟,φ⟩_𝒟(Ω)=st( u φ^∗ dx).


Moreover, with some abuse of notation, we will write also that [u]_𝒟∈ L^2(Ω), [u]_𝒟∈ V(Ω),
etc. meaning that the distribution [u]_𝒟
can be identified with a function f in L^2(Ω), V(Ω),
etc. By our construction, this is equivalent to say that f^∗∈[u]_𝒟.
So, in this case, we have that ∀φ∈𝒟(Ω),

    ⟨[u]_𝒟,φ⟩ _𝒟(Ω)=st(∫^∗u φ^∗ dx)=st(∫^∗f^∗φ^∗dx)=∫ f φ dx.



Since an ultrafunction u: Ω^*→ℝ^*
is univocally determined by its value in Γ, we may think of
the ultrafunction as being defined only on Γ and to denote
them by V_Λ(Γ); the set V_Λ(Γ) is
an algebra which extends the algebra of continuous functions 𝒞(Ω)
if it is equipped with the pointwise product. 

Moreover, we recall that, by a well known theorem of Schwartz, any
tempered distribution can be represented as ∂^αf,
where α is a multi-index and f is a continuous function.
If we identify T=∂^αf with the ultrafunction D^αf°,
we have that the set of tempered distributions 𝒮' is
contained in V_Λ(Γ). However the Schwartz impossibility
theorem (see introduction) is not violated since (V_Λ(Γ),+, · , D)
is not a differential algebra, since the Leibnitz rule does not hold
for some couple of ultrafunctions.




§ CONSTRUCTION OF THE CACCIOPPOLI SPACE OF ULTRAFUNCTIONS


In this section we will prove the existence of Caccioppoli spaces
of ultrafunctions (see Definition <ref>) by an explicit
construction.



 §.§ Construction of the space V_Λ(Ω)


In this section we will construct a space of ultrafunctions V_Λ(Ω)
and in the next section we will equip it with a σ-basis in
such a way that V_Λ(Ω) becomes a Caccioppoli space
of ultrafunctions according to Definition <ref>.

Given a family of open sets ℜ_0 , we say that a family
of open sets 𝔅={ E_k} _k∈ K is a
basis for ℜ_0 if 


  * ∀ k≠ h, E_k∩ E_h=∅; 

  * ∀ A∈ℜ_0, there is a set of indices K_E⊂ K
such that

    A=int(⋃_k∈ K_EE_k);



  * 𝔅 is the smallest family of sets which satisfies the
above properties.

We we will refer to the family ℜ of all the
open sets which can be written by the espression (<ref>)
as to the family generated by ℜ_0. 

Let us verify that 

For any finite family of special Caccioppoli sets ℭ_0,
there exists a basis 𝔅 whose elements are special Caccioppoli
sets. Moreover also the set ℭ generated by ℭ_0
consists of special Caccioppoli sets.


For any x∈Ω, we set 

    E_x=⋂{ A∈ℭ_0 | x∈ A} .

We claim that { E_x} _x∈Ω is a basis. Since
ℭ_0 is a finite family, then also { E_x} _x∈Ω
is a finite family and hence there is a finite set of indices K
such that 𝔅={ E_k} _k∈ K. Now it
is easy to prove that 𝔅 is a basis and it consists of
special Caccioppoli sets. Also the last statement is trivial.

We set 

    ℭ_0,λ(Ω):=λ∩ℭ(Ω),

and we denote by 𝔅_λ(Ω) and ℭ_λ(Ω)
the relative basis and the generated family which exist by the previous
lemma.


Now set

    ℭ_Λ(Ω)=lim_λ↑Λℭ_λ(Ω), 𝔅_Λ(Ω)=lim_λ↑Λ𝔅_λ(Ω).



The following properties hold true



  * ℭ_Λ(Ω) and 𝔅_Λ(Ω)
are hyperfinite;

  * ℭ(Ω)^σ⊂ℭ_Λ(Ω)⊂ℭ(Ω)^*;

  * if E∈ℭ_Λ(Ω), then

    θ_E=∑_Q∈ K(E)θ_Q(x),

 ℭ_Λ(Ω)^* ℭ_Λ(Ω)



It follows trivially by the construction.

The next lemma is a basic step for the construction of the space V_Λ(Ω).

For any Q∈𝔅_Λ(Ω)
there exists a set 
and a family of functions


  * Ω⊂Ξ⊂Ω^*;

  * if Q,R∈𝔅_Λ(Ω) and Q≠ R, then 

  * if a∈Ξ(Q), the

  * 

  * ζ_a≥0;

  * 

    ζ_a(a)=1.




We set

    r(λ)=1/3min{ d(x,y) | x,y∈λ∩Ω} ,

and we denote by ρ a smooth bell shaped function having support
in B_1(0); then the functions ρ(x-a_λ/r(λ)), a_λ∈λ∩Ω
have disjoint support. We set

    Ξ:={lim_λ↑Λa_λ | a_λ∈λ∩Ω} ,

so that Ω⊂Ξ⊂Ω^* and we divide all points
a∈Ξ, among sets Ξ(Q), Q∈𝔅_Λ, in
such a way that

- if a∈ Q then a∈Ξ(Q);

- if a∈∂ Q_1∩...∩∂ Q_l there exists a
unique Q_j (j≤ l) such that a∈Ξ(Q_j).

With this construction, claims 1. and 2. are trivially satisfied.
Now, for any a∈Ξ(Q), set 

    ρ_a(x):=lim_λ↑Λρ(x-a_λ/r(λ)),

and

    ζ_a(x):=ρ_a(x)θ_Q(x)/ρ_a(a)θ_Q(a).

It is easy to check that the functions ζ_a satisfy 3.,4.,5.
and 6. 

We set

    V_Λ^1(Ω)=span({ζ_a} _a∈Ξ)+lim_λ↑Λ(λ∩𝒞^1(Ω)),

and

    V_Λ^1(Q)={ uθ_Q | u∈ V_Λ^1(Ω)} ;

so we have that, for any a∈Ξ(Q), ζ_a∈ V_Λ^1(Q).
Also, we set


    V_Λ^0(Ω)=Span({ f, ∂_if, fg, g∂_if  |  f,g∈ V_Λ^1(Ω),  i=1,...,N} +lim_λ↑Λ(λ∩𝒞(Ω)))

and

    V_Λ^0(Q)={ uθ_Q | u∈ V_Λ^0(Ω)} .

 

Finally, we can define the V_Λ(Ω)
as follows:

    V_Λ(Ω)=Q∈𝔅_Λ(Ω)⊕V_Λ^0(Q).

Namely, if u∈ V_Λ(Ω), then

    u(x)=∑_Q∈𝔅_Λ(Ω)u_Q(x)θ_Q(x),

with u_Q∈ V_Λ^0(Ω).



 §.§ The σ-basis 


In this section, we will introduce a σ-basis in such a way
that V_Λ(Ω) becomes a Caccioppoli space of ultrafunctions,
according to Definition <ref>.

 There exists a σ-basis V_Λ(Ω), {σ_a(x)} _a∈Γ,
such that



  * Ω⊂Γ⊂Ω^*;

  * Γ=⋃__Q∈𝔅_Λ(Ω)Q_Γ,
where Q∩Γ⊂ Q_Γ⊂Q∩Γ,
and Q_Γ∩ R_Γ=∅  for  Q≠ R; 

  * {σ_a(x)} _a∈ Q_Γ is a σ-basis
V_Λ^0(Q).



First we introduce in V_Λ(Ω) the following
scalar product:

    ⟨ u,v⟩ =∫^*uv dx.


For any Q∈𝔅_Λ(Ω) we set

    Z(Q)={∑_a∈Ξ(Q)γ_aζ_a(x) | γ_a∈ℝ^*} ,

where Ξ(Q) and the functions{ζ_a} _a∈Ξ
are defined in Lemma <ref>.

If we set 

    𝔡_a(x)=ζ_a(x)/∫^*|ζ_a(x)|^2 dx

we have that

    {𝔡_a(x)} _a∈Ξ(Q)

is a δ-basis for Z(Q)⊂ V_Λ^0(Q) (with respect
to the scalar product (<ref>)). In fact, if u∈ Z(Q),
then u(x)=∑_b∈Ξ(Q)u(b)ζ_b(x), and hence, by Lemma
<ref>, it follows that

    ∫^*u(x)𝔡_a(x) dx     =∫^*∑_b∈Ξ(Q)u(b)ζ_b(x)𝔡_a(x) dx
        =∑_b∈Ξ(Q)u(b)∫^*ζ_b(x)𝔡_a(x) dx
        =∑_b∈Ξ(Q)u(b)∫^*ζ_b(x)(ζ_a(x)/∫^*|ζ_a(x)|^2 dx )dx
        =∑_b∈Ξ(Q)u(b)δ_ab=u(a).


Next, we want to complete this basis and to get a δ-basis
for V_Λ^0(Q). To this aim, we take an orthonormal basis
{ e_k(x)} of Z(Q)^⊥ where Z(Q)^⊥
is the orthogonal complement of Z(Q) in V_Λ^0(Q) (with
respect to the scalar product (<ref>)). For every a∈ Q\Ξ,
set

    𝔡_a(x)=∑_ke_k(a)e_k(x);

notice that this definition is not in contradiction with (<ref>)
since in the latter a∈Ξ.

For every v∈ Z(Q)^⊥, we have that

    ∫^*v(x)𝔡_a(x) dx=v(a);

in fact

    ∫^*v(x)𝔡_a(x) dx     =∫^*(∑_kv_ke_k(x))(∑_he_h(a)e_h(x)) dx=∑_k,hv_ke_h(a)∫^*e_k(x)e_h(x) dx
        =∑_k,hv_ke_h(a)δ_hk=∑_kv_ke_k(a)=v(a).

It is not difficult to realize that {𝔡_a(x)} _a∈ Q\Ξ
generates all Z(Q)^⊥ and hence we can select a set Ξ^⋆(Q)⊂ Q\Ξ
such that {𝔡_a(x)} _a∈Ξ^⋆(Q)
is a basis for Z(Q)^⊥. Taking 

    Q_Γ=Ξ^⋆(Q)∪Ξ(Q),

we have that {𝔡_a(x)} _a∈ Q_Γ
is a basis for V_Λ^0(Q).

Now let {σ_a(x)} _a∈ Q_Γ denote
the dual basis of {𝔡_a(x)} _a∈ Q_Γ
namely a basis such that, ∀ a,b∈ Q_Γ,

    ∫^*σ_a(x)𝔡_b(x)dx=δ_ab.


Clearly it is a σ-basis for V_Λ^0(Q). In fact,
if u∈ V_Λ^0(Q), we have that

    u(x)=∑_a∈ Q_Γ[∫^*u(t)𝔡_a(t) dt]σ_a(x)=∑_a∈ Q_Γu(a)σ_a(x).

Notice that if a∈Ξ(Q), then σ_a(x)=ζ_a(x).
The conclusion follows taking 

    Γ:=⋃_Q∈𝔅_Λ(Ω)Q_Γ.


By the above theorem, the following corollary follows straightforward.

 Caccioppoli
space of ultrafunctions

If E∈ℭ_Λ(Ω) (see (<ref>)),
we set

    E_Γ=⋃_Q∈𝔅_Λ(Ω),Q⊂ EQ_Γ

If, for any internal set A, we define

    _Au(x)dx=∑_a∈Γ∩ Au(a)η_a

then, we have the following result:

If uθ_E∈ V_Λ(Ω) and
E∈ℭ_Λ(Ω), then

    _E_Γu(x)dx=∫_E^*u(x)dx= u(x)θ_E(x)dx.



We have that

    _E_Γu(x)dx     =∑_a∈ E_Γu(a)η_a=∫^*∑_a∈ E_Γu(a)σ_a(x)dx
        =∫^*∑_Q⊂ E ∑_a∈ Q_Γu(a)σ_a(x)dx

Since uθ_E∈ V_Λ(Ω), then, by (<ref>),
we can write 

    u(x)θ_E(x)=∑_Q⊂ Eu_Q(x)θ_Q(x).


By Th. <ref>, 3., u_Q(x)θ_Q(x)=∑_a∈ Q_Γu(a)σ_a(x)∈ V_Λ^0(Q).
Then by (<ref>)


    _E_Γu(x)dx     =∫^*∑_Q⊂ E u_Q(x)θ_Q(x)dx=∑_Q⊂ E ∫^*u_Q(x)θ_Q(x)dx.

By this equation and the fact that ∫^*u_Q(x)θ_Q(x)dx=∫_Q^*u(x)dx,
it follows that


    _E_Γu(x)dx=∑_Q⊂ E ∫_Q^*u(x)dx=∫_E^*u(x)dx.

Moreover, since u_Qθ_Q∈ V_Λ^0(Q)⊂ V_Λ(Ω),

    ∫^*u_Q(x)θ_Q(x)dx= u_Q(x)θ_Q(x)dx,

by (<ref>) we have

    _E_Γu(x)dx=∑_Q⊂ E u_Q(x)θ_Q(x)dx=∑_Q⊂ Eu_Q(x)θ_Q(x)dx= u(x)θ_E(x)dx.





 §.§ Construction of the generalized derivative


Next we construct a generalized derivative on V_Λ(Ω).

We set

    U_Λ^1=Q∈𝔅_Λ(Ω)⊕V_Λ^1(Q),

 and 

    U_Λ^0=(U_Λ^1)^⊥,

will denote the the orthogonal complement of U_Λ^1 in
V_Λ(Ω). According to this decomposition, V_Λ(Ω)=U_Λ^1⊕ U_Λ^0
and we can define the following orthogonal projectors

    P_i : V_Λ(Ω)→ U_Λ^i,  i=0,1  ,

hence, any ultrafunction u∈ V_Λ(Ω) has the following
orthogonal splitting: u=u_1+u_0 where u_i=P_iu.

Now we are able to define the generalized partial derivative for u∈ V_Λ^1(Ω).

We define the generalized partial derivative

    D_i:U_Λ^1(Ω)→ V_Λ(Ω),

 as follows: 

    D_iuv dx=∑_Q∈𝔅_Λ(Ω)∂_i^*u_Qv_Qθ_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(u_Q-u_R)v_Q (𝐞_i·𝐧_Q) dS,

where 

    𝔜(Q)={ R∈𝔅_Λ(Ω)∪{ Q_∞} | Q≠ R,  ∂ Q∩∂ R≠∅}

with 

    Q_∞=Ω^*∖⋃_Q∈𝔅_Λ(Ω)Q,

Moreover, if u=u_1+u_0∈ U_Λ^1⊕ U_Λ^0=V_Λ(Ω),
we set

    D_iu=D_iu_1-(D_iP_1)^†u_0,

where, for any linear operator L, L^† denotes the adjoint
operator.




Notice that, if u,v∈ U_Λ^1, by Th. <ref>,2.,
we have

    D_iuv dx=∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(u_Q-u_R)v_Q (𝐞_i·𝐧_Q) dS.

In fact, if u, v∈ U_Λ^1, then u_Q, v_Q∈ V_Λ^1(Ω)
and hence, by (<ref>), ∂_i^*u_Qv_Q∈ V_Λ^0(Ω)
and so

    ∂_i^*u_Qv_Qθ_Q dx=∫_Q^*∂_i^*u_Qv_Q dx.





The operator D_i: V_Λ(Ω)→ V_Λ(Ω),
given by Definition <ref>, satisfies the requests
I, II, and III of Definition <ref>. 


Let us prove property I. If uθ_Q,vθ_R∈ V_Λ(Ω)
and Q∩R=∅, by Definition <ref>,

    D_i(uθ_Q)vθ_R dx=0.

 Set

    δ:=max{ diam(Q) | Q∈𝔅_Λ(Ω)} .

If q∈ Q and r∈ R, then

    |q-r|>2δ⇒Q∩R=∅,

so, if σ_q∈ V_Λ^0(Q), and r∈ R, then

    |q-r|>2δ⇒Q∩R=∅,

and hence, if we set ε_0>3δ,

    ⋃{ R∈𝔅_Λ(Ω) | Q∩R≠∅}⊂ B_ε_0(q).

Since σ_q∈ V_Λ^0(Q),

    𝔰𝔲𝔭𝔭(D_iσ_q)⊂⋃{ R∈𝔅_Λ(Ω) | Q∩R≠∅}⊂ B_ε_0(q).


We prove property II. If u∈[C^1(Ω)]^*∩ V_Λ(Ω),
then u=∑__Q∈𝔅_Λ(Ω)uθ_Q,
and hence ∀ x∈∂ Q∩∂ R, u_Q(x)-u_R(x)=u(x)-u(x)=0.
Then, by (<ref>), we have that, ∀ v∈ V_Λ(Ω) 


    D_iuv dx     =∑_Q∈𝔅_Λ(Ω)∂_i^*uv_Qθ_Q dx=∂_i^*u(∑_Q∈𝔅_Λ(Ω)v_Qθ_Q) dx
        =∂_i^*uv dx.

The conclusion follows from the arbitrariness of v.

Next let us prove property III. First we prove this property if u,v∈ U_Λ^1.
By (<ref>), we have that

    D_iuv dx     =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Qv_Q (𝐞_i·𝐧_Q) dS
        +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Rv_Q (𝐞_i·𝐧_Q) dS
        =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS
        +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Rv_Q (𝐞_i·𝐧_Q) dS.

Next we will compute uD_iv dx and we will show that it
is equal to - D_iuv dx. So we replace u with v,
in the above equality and we get

    ∑_Q∈𝔅_Λ(Ω)∫_Q^*uD_iv dx     =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*v_Qu_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS
        +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS.

Now, we compute ∑__Q∈𝔅_Λ(Ω)__Q^*∂_i^*v_Qu_Q dx
and the last term of the above expression separately. We have that


    ∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*v_Qu_Qdx     =-∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Qdx+∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS.

Moreover, the last term in (<ref>), changing the order on which
the terms are added, becomes

    ∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS=∑_R∈𝔅_Λ(Ω)∑_Q∈𝔜(R)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS.

In the right hand side we can change the name of the variables Q
and R:

    ∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS     =∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_R) dS
        =-∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_Q) dS.

In the last step we have used the fact that x∈∂ Q∩∂ R⇒𝐧_R(x)=-𝐧_Q(x).
Replacing (<ref>) and (<ref>) in (<ref>)
we get 

    ∑_Q∈𝔅_Λ(Ω)∫_Q^*uD_iv dx     =-∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Qdx+1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS
        -1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_Q) dS.


Comparing (<ref>) and the above equation, we get that


    ∀ u,v∈ U_Λ^1,   D_iuv dx=- uD_iv dx.


Let us prove property III in the general case. We have that 

    D_iu=D_iu_1-(D_iP_1)^†u_0

hence


    D_iuv dx     = D_iu_1v dx-(D_iP_1)^†u_0v dx
        = D_iu_1v_1 dx+ D_iu_1v_0 dx- u_0D_iP_1v dx
        = D_iu_1v_1 dx+ D_iu_1v_0 dx- u_0D_iv_1 dx.

Now, replacing u with v and applying property (<ref>)
for u_1,v_1∈ U_Λ^1, we get 

    uD_iv dx     = u_1D_iv_1 dx- D_iu_1v_0 dx+ u_0D_iv_1 dx
        =- D_iu_1v_1 dx- D_iu_1v_0 dx+ u_0D_iv_1 dx.

Comparing the above equation with (<ref>) we get that

    D_iuv dx=- uD_iv dx.


Before proving property IV we need the following lemma:

The following identity holds true: ∀ E∈ℭ_Λ(Ω),
∀ u∈ V_Λ(Ω)∩[C^1(Ω)]^*
and ∀ v∈ V_Λ(Ω), 

    D_i(uθ_E)v dx=∂_iuvθ_E dx-∫_∂ E^*uv (𝐞_i·𝐧_E) dS.



We can write

    uθ_E=∑_Q∈𝔅_Λ(Ω)h_Quθ_Q,

where

    h_Q=
    1     if Q⊂ E
    
    0     if Q⊈ E

Then, we have that 

    h_Qu-h_Ru=u,

 if and only if,

    R∈𝔜_Q,E^+:={ R∈𝔅_Λ(Ω)∪{ Q_∞} | ∂ R∩∂ E≠∅, Q⊂ E, R⊂Ω∖ E} .

 Moreover, we have that,

    h_Qu-h_Ru=-u,

if and only if,

    R∈𝔜_Q,E^-:={ R∈𝔅_Λ(Ω)∪{ Q_∞} | ∂ R∩∂ E≠∅, Q⊂Ω∖ E, R⊂ E} .

Otherwise, we have that 

    h_Qu-h_Ru=0    or    ∂ R∩∂ E=∅.

Then, by Theorem <ref>,

    D_i(u_Eθ_E)v dx     =∑_Q∈𝔅_Λ(Ω)∂_i^*(h_Qu)vθ_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(h_Qu-h_Ru)v (𝐞_i·𝐧_Q) dS
        =∑_Q∈𝔅_Λ(Ω)h_Q_Q_Γ∂_i^*uvdx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜_Q,E^+∫_∂ Q∩∂ R^*uv (𝐞_i·𝐧_Q) dS
        +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜_Q,E^-∫_∂ Q∩∂ R^*uv (𝐞_i·𝐧_Q) dS
        =_E_Γ∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω),Q⊂ E∫_∂ Q∩∂ E^*uv (𝐞_i·𝐧_E) dS
        +1/2∑_Q∈𝔅_Λ(Ω), Q⊂Ω∖ E∫_∂ Q∩∂ E^*uv (𝐞_i·(-𝐧_E)) dS
        =∂_i^*uvθ_E dx-1/2∫_∂ E^*uv (𝐞_i·𝐧_E) dS+1/2∫_∂ E^*uv (𝐞_i·(-𝐧_E)) dS
        =∂_i^*uvθ_E dx-∫_∂ E^*uv (𝐞_i·𝐧_E) dS.

 Then (<ref>) holds true.


The operator D_i: V_Λ(Ω)→ V_Λ(Ω)
given by Definition <ref> satisfies the request IV
of Definition <ref>. 


The result follows straighforward from (<ref>) just taking
u=1.




§ SOME EXAMPLES


We present a general minimization result and two very basic examples
which can be analyzed in the framework of ultrafunctions. We have
chosen these examples for their simplicity and also because we can
give explicit solutions. 



 §.§ A minimization result


In this section we will consider a minimization problem. Let Ω
be an open bounded set in ℝ^N and let Ξ⊂∂Ω
be any nonempty portion of the boundary. We consider the following
problem: minimize 

    J(u)=∫_Ω[1/2a(u)|∇ u(x)|^p+f(x,u)]dx,  p>1

in the set

    𝒞^1(Ω)∩𝒞_0(Ω∪Ξ).

We make the following assumptions:


  * a(u)≥0 and a(u)≥ k>0 for u sufficiently large;

  * a(u) is lower semicontinuous;

  * f(x,u) is a lower semicontinuous function in u, measurable in
x, such that |f(x,u)|≤ M|u|^q, with 0<q<p and M∈ℝ^+.

Clearly, the above assumptions are not sufficient to guarantee the
existence of a solution, not even in a Sobolev space. We refer to
<cit.> for a survey of this problem in the framework of Sobolev
spaces. On the other hand, we have selected this problem since it
can be solved in the framework of the ultrafunctions. 

More exactly, this problem becomes: find an ultrafunction u∈ V_Λ(Ω)
which vanishes on Ξ^* and minimizes

    J°(u):=_Ω[1/2a^*(u) |Du(x)|^p-f^*(x,u)]dx,   p>1.


We have the following result:

If assumptions 1,2,3 are satisfied, then the functional
J°(u) has a minimizer in the space

    { v∈ V_Λ(Ω) | ∀ x∈Ξ^*, v(x)=0 } .

Moreover, if J(u) has a minimizer w in V_Λ(Ω),
then u=w°.


The proof is based on a standard approximation by finite dimensional
spaces. Let us observe that, for each finite dimensional space V_λ,
we can consider the approximate problem: find u_λ∈ V_λ
such that


    J°(u_λ)=min_v_λ∈ V_λJ°(v_λ).

The above minimization problem has a solution, being the functional
coercive, due to the hypotheses on a(·) and the fact that p>q.
If we take a miminizing sequence u_λ^n∈ V_λ,
then we can extract a subsequence weakly converging to some u_λ∈ V_λ.
By observing that in finite dimensional spaces all norms are equivalent,
it follows also that u_λ^n→ u_λ pointwise.
Then, by the lower-semicontinuity of a and f, it follows that
the pointwise limit satisfies


    J°(u_λ)≤lim inf J°(u_λ^n).

Next, we use the very general properties of Λ-limits, as
introduced in Section <ref>. We set

    u:=lim_λ↑Λu_λ.

Then, taking a generic v:=lim_λ↑Λv_λ,
from the inequality J°(u_λ)≤ J°(v_λ),
we get

    J°(u)≤ J°(v)    ∀ v∈ V_Λ(Ω).

The last statement is trivial.

Clearly, under this generality, the solution u could be very wild;
however, we can state a regularization result which allows the comparison
with variational and classical solutions.

Let the assumptions of the above theorem hold. If

    ℋ^N-1(Ξ)>0

and there exists ν∈ℝ such that

    a(u)≥ν>0,

then, the minimizer has the following form

    u=w^∘+ψ,

where w∈ H^1,p(Ω) and ψ is null in the sense of
distributions, namely

    ∀φ∈𝒟(Ω),  ψφ^*dx∼0.

In this case 

    J°(u)∼v∈ V(Ω)infJ°(v)

with V(Ω) as in Definition <ref>. Moreover, if in
addition a(u)<M, with M∈ℝ, we have that 

    ‖ψ‖ _H^1,p(Ω)∼0

and J°(u)∼ J(w). Finally, if u=w^∘ and J°(u)=J(w). 


Under the above hypotheses the minimization problem has an additional
a priori estimates in H^1,p(Ω), due to the fact that a(·)
is bounded away from zero. Moreover, the fact that the function vanishes
on a non trascurable (N-1)-dimensional part of the boundary, shows
that the generalized Poincar� inequality holds true. Hence, by Proposition
<ref>, the approximating net { u_λ}
has a subnet { u_n} such that

    u_n→ u  weakly in   H^1,p(Ω).

This proves the first statement, since obviously, ψ:=u-w^∘
vanishes in the sense of distributions. In this case, in general the
minimum is not achieved in V(Ω) and hence J°(w^∘+ψ)<J(w).

Next, if a(·) is bounded also from above, by classical results
of semicontinuity of De Giorgi (see Boccardo <cit.> Section
9, Thm. 9.3) J is weakly l.s.c. Thus u is a minimizer and, by
well known results, u_n→ u strongly in H^1,p(Ω).
This implies, by Proposition <ref>, that u∼ w^∘,
and hence ψ is infinitesimal in H^1,p(Ω), proving
the second part. 

Finally, if the minimizer is a function w∈ H^1,p(Ω)∩𝒞(Ω)⊂ V_Λ(Ω),
we have that u_λ=w° eventually; then 


    ‖ψ‖ _H^1,p(Ω)=0.





 §.§ The Poisson problem in ℝ^2


Now we cosider this very classical problem:

    - u=φ(x),    φ∈𝒟(ℝ^N).


If N≥3, the solution is given by 

    φ(x)∗|x|^-N+2/(N-2)ω_N

and it can be characterized in several ways. 

First of all, it is the only solution the Schwartz space 𝒮'
of tempered distributions obtained via the equation 

    u(ξ)=φ(ξ)/|ξ|^2

where T denotes the Fourier tranform of T.

Moreover, it is the minimizer of the Dirichlet integral

    J(u)=∫[1/2|∇ u(x)|^2-φ(x)u(x)]dx

in the space 𝒟^1,2(ℝ^N) which is defined
as the completion of 𝒞^1(ℝ^N)
with respect to the Dirichlet norm

    ‖ u‖ =√(∫|∇ u(x)|^2dx).


Each of these characterizations provides a different method to prove
its existence.

The situation is completely different when N=2. In this case, it
is well known that the fundamental class of solutions is given by

    2π·φ(x)∗ log|x|,

 however none of the previus characterization makes sense. In fact,
we cannot use equation (<ref>), since 1/|ξ|^2∉ L_loc^1(ℝ^2)
and hence 1/|ξ|^2 does not define a tempered distribution.
Also, the space 𝒟^1,2(ℝ^2) is not an Hilbert
space and the functional J(u) is not bounded from below in 𝒟^1,2(ℝ^2). 

On the contrary, using the theory of ultrafunctions, we can treat
equation (<ref>) independently of the dimension. 

First of all, we recall that in equation (<ref>) with
N≥3, the boundary conditions are replaced by the condition u∈𝒟^1,2(ℝ^N).
This is a sort of Dirichlet boundary condition. In the theory of ultrafunctions
it is not necessary to replace the Dirichlet boundary condition with
such a trick. In fact we can reformulate the problem in the following
way: find u∈ V_Λ(B_R) such that 

    -°u     =φ°(x)   in  B_R
    
    u     =0           on  ∂ B_R

where ° is the “generalized” Laplacian
defined in Section <ref> and R is an infinite
number such that χ__B_R∈ V_Λ(ℝ^N).
[Such an R exists by overspilling (see e.g. <cit.>);
in fact for any r∈ℝ, χ__B_r∈ V_Λ(ℝ^N). ]

Clearly, the solutions of the above problem are the minimizers of
the Dirichlet integral

    J°(u)=[1/2|Du(x)|^2-φ°(x)u(x)]dx

in the space u∈ V_Λ(B_R), with the Dirichlet boundary
condition. Notice that, in the case of ultrafunctions, the problem
has the same structure independently of N. In order to prove the
existence, we can use Theorem <ref>[The fact that Ω is a standard set while B_R is an internal
set does not change the proof.]. The fact that J°(u) may assume infinite values
does not change the structure of the problem and shows the utility
of the use of infinite quantities. The relation between the classical
solution w and the ultrafunction u is given by 

    u=w°+ψ

with

    St_𝒟'ψ=0.


Some people might be disappointed that u depends on R and it
is not a standard function; if this is the case it is sufficient to
take

    w=St_𝒟'u

and call w the standard solution of the Poisson problem with
Dirichlet boundary condition at ∞. In this way we get the
usual fundamental class of solutions and they can be characterized
in the usual way also in the case N=2. Concluding, in the framework
of ultrafunctions, the Poisson problem with Dirichlet boundary condition
is the same problem independently of the space dimension
and and it is very similar to the same problem when R is finite. 

This fact proves that the use of infinite numbers is an advantage
which people should not ignore.



 §.§ An explicit example


If the assumptions of Theorem <ref> do not hold true, the
solution could not be related to any standard object. For example,
if ℋ^N-1(Ξ)=0 and f(x,u)>k|u|^s,  (p<N,  k>0,  0<s<q),
the generalized solution u(x) takes infinite values for every x∈Ω.
However, there are cases in which u(x) can be identified with a
standard and meaningful function, but the minimization problem makes
no sense in the usual mathematics. In the example which we will present
here, we deal with a functional which might very well represent a
physical model, even if the explicit solution cannot be interpreted
in a standard world, since it involves the square of a measure (namely
δ^2).

Let us consider, for γ>0, the one dimensional variational
problem of finding the minimum of the functional 

    J(u)=∫_0^11/2a(u)|u'(x)|^2-γ u(x) dx

among the functions such that u(0)=0. In particular we are interested
in the case in which a is the following degenerate function

    a(s)={    1         if s∈(-∞,1)∪(2,+∞),
        0         if s∈[1,2].
    .

Formally, the Euler equation, if u∉[1,2], is 

    u”(x)=-γ.

We recall that, by standard arguments, 

    u(1)≠1⇒ u'(1)=0.

Hence, if γ<2, the solution is explicitly computed 

    u(x)=γ/2(2x-x^2),

since it turns out that 0≤ u(x)<1 for all x∈(0,1) and then
the degeneracy does not take place.

If γ>2, we see that the solution does not live in H^1(0,1),
hence the problem has not a “classical” weak solution. More exactly
we have the following result: 

If 

    u(x)={   1/2(2γ x-γ x^2)      0<x<ξ
       1/2(-γ x^2+2γ x+2)     ξ<x<1
    .

where Figure 2).


First, we show that the generalized solution has at most one discontinuity.
In fact, for γ>2 the solution satisfies u(ξ)=1, for some
0<ξ<1, and at that point the classical Euler equations are not
anymore valid. On the other hand, where u>2, the solution satisfies
a regular problem, hence we are in the situation of having at least
the following possible candidate as solution with a jump at ξ=γ-√(γ^2-2γ)/γ=1-√(1-2/γ) 
and a discontinuity of derivatives at some ξ<η<1. 

In the specific case, we have (see Figure 1)

    u(x)={   1/2(2γ x-γ x^2)      0<x<ξ
        2     ξ<x<η
       γη^2/2-γη-γ x^2/2+γ x+2     η<x<1.
    .

We now show that this is not possible because the functional takes
a lower value on the solution with only a jump at x=ξ. In fact,
if we consider the function ũ(x) defined as follows

    ũ(x)={   1/2(2γ x-γ x^2)      0<x<ξ
       1/2(-γ x^2+2γ x+2)     ξ<x<1
    .

we observe that u=ũ in [0,ξ], while u'=ũ'=γ(1-x)
for all x∉[ξ,η] and, by explicit computations, we have

    J(ũ)-J(u)=γ^2[-η/2+η^2/2-η^3/6+ξ/2-ξ^2/2+ξ^3/6]=γ^2(Φ(η)-Φ(ξ))<0,  ξ<η

where Φ(s)=-s/2+s^2/2-s^3/6 is strictly
decreasing, since Φ'(s)=-1/2(s-1)^2≤0. 


Actually, the solution is the one shown in Figure 2. Next we show
that there exists a unique point ξ such that the minimum is attained.
We write the functional J(u), on a generic solution with a single
jump from the value u=1 to the value u=2 at the point 0<ξ<1
and such that the Euler equation is satisfied before and after ξ.
We obtain the following value for the functional (in terms of the
point ξ) 

    J(u)=F(ξ)=γ^2ξ^3/8-γ^2ξ^2/2+γ^2ξ/2-γ^2/6+3γξ/2-2γ+1/2ξ.

We observe that, ∀ γ>2 

    F(0^+)=+∞    and     F(1)=-γ^2/24-γ/2+1/2<0.

To study the behavior of F(ξ) one has to solve some fourth order
equations (this could be possible in an explicit but cumbersome way),
so we prefer to make a qualitative study. We evaluate 

    F'(ξ)     =3γ^2ξ^2/8-γ^2ξ+γ^2/2+3γ/2-1/2ξ^2,
    
    F”(ξ)     =3γ^2ξ/4-γ^2+1/ξ^3,
    
    F”'(ξ)     =3γ^2/4-3/ξ^4,

hence we have that 

    F”'(ξ)<0   if and only if   0<ξ<√(2/γ)<1.

Consequently the function F”(ξ), which nevertheless satisfies
∀ γ>2 

    F”(0^+)=+∞,     F”(1)=1-γ^2/4<0,

has a unique negative minimum at the point √(2/γ).


From this, we deduce that there exists one and only one point 0<ξ_0<√(2/γ)
such that 

    F”(ξ)>0        0<ξ<ξ_0
        F”(ξ)<0        ξ_0<ξ≤1.

From the sign of F” we get that F' is strictly increasing in
(0,ξ_0) and decreasing in (ξ_0,1). Next ∀ γ>2

    F'(0^+)=-∞,     F'(1)=-γ^2/8+3γ/2-1/2

hence, in the case that 

    -γ^2/8+3γ/2-1/2>0   that is γ<2(3+2√(2))∼11.656...,

then F' has a single zero ξ_1∈(0,ξ_0) and, being a
change of sign, ξ_1 is a point of absolute minimum for F(ξ).

If γ≥2(3+2√(2)) the above argument fails.



In this case we can observe that 

    F'(1/γ)=γ/2+3/8>0,

hence F'(1), which is negative at ξ=1 and near ξ=0 vanishes
exactly two times, at the point ξ_1, which is a point of local
minimum and at another point ξ_2>ξ_1, which is a point
of local maximum. Hence, to find the absolute minimum, we have to
compare the value of F(ξ_1) with that of F(1).

In particular, we have that ξ_1<√(2/γ) hence, we can
show that the minimum is not at ξ=1 simply by observing that
we can find at least a point where F(ξ)<F(1) and this point is
√(2/γ). In fact 

    M(γ):=F(√(2/γ))-F(1)=γ^3/2/√(2)-γ^2/8-5γ/2+2√(2)√(γ)-1/2≤0.

In particular M(2)=0 and 

    M'(γ)=1/4(-γ+3√(2)√(γ)+4√(2)/√(γ)-10)<0.

This follows since by the substituting √(γ)→χ, we
have to control the sign of the cubic 

    M̃(χ)=-χ^3+3√(2)χ^2-10χ+4√(2)

which is negative for all χ≥1, since M̃(1)=-11+7√(2),
while 

    M̃'(χ)=-3χ^2+6√(2)χ-10,

is a parabola with negative minimum.


It could be interesting to study this problem in dimension bigger
that one, namely, to minimize

    J°(u)=_Ω(1/2a(u)|Du(x)|^2-γ u(x)) dx

in the set 

    { v∈ V_Λ(Ω) | ∀ x∈Ξ^*, u(x)=0 }

and in particular to investigate the structure of the singular set
of u, both in the general case and in some particular situations
in which it is possible to find explicit solutions (e.g. Ω=B_R(0)).

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