New trends in free boundary problems
Serena Dipierro Aram Karakhanyan Enrico Valdinoci



We present a series of recent results
on some new classes of free boundary problems.
Differently from the classical literature, the problems
considered have either a “nonlocal”
feature (e.g., the interaction or/and the interfacial
energy may
depend on global quantities) or a “nonlinear”
flavor (namely, the total energy is the nonlinear
superposition of energy components, thus losing
the standard additivity and scale invariances of the problem).

The complete proofs and the full details of the results
presented here are given in <cit.>.



[2010]35R35, 35R11.

Multi-Year X-ray Variations of Iron-K and Continuum Emissions 
 in the Young Supernova Remnant Cassiopeia A
    Toshiki Sato1,2, Yoshitomo Maeda2,
   Aya Bamba3,4,
   Satoru Katsuda5,
   Yutaka Ohira3,
   Ryo Yamazaki3,
   Kuniaki Masai1,
   Hironori Matsumoto6,
   Makoto Sawada3,
   Yukikatsu Terada7,
   John P. Hughes8
   and Manabu Ishida2
   
    January 2017
================================================================================================================================================================================================================================================




Para Don Ireneo, por supuesto.







§ INTRODUCTION


In this survey, we would like to present some
recent research directions in the study of variational
problems whose minimizers naturally exhibit the formation
of free boundaries. Differently than the cases
considered in most of the existing literature, the problems
that we present here are either nonlinear (in the sense
that the energy functional is the nonlinear superposition
of classical energy contributions) or nonlocal
(in the sense that some of the energy contributions
involve objects that depend on the global geometry
of the system).

In these settings, the problems typically show new features
and additional difficulties with respect to the classical cases.
In particular, as we will discuss in further details:
the regularity theory is more complicated,
there is a lack of scale invariance for some problems,
the natural scaling properties of the energy may not
be compatible with the optimal regularity,
the condition at the free boundary may be of nonlocal
or nonlinear type and involve the global behavior of
the solution itself, and 
some problems may exhibit a variational instability
(e.g., minimizers in large domains and in small domains
may dramatically differ the ones from the others).

We will also discuss
how the classical free boundary problems in <cit.>
are recovered either as limit problems
or after a blow-up, under appropriate structural
conditions on the energy functional.

We recall the classical free boundary problems
of <cit.>
in Section <ref>.

The results concerning nonlocal free boundary problems
will be presented in Section <ref>,
while the case of 
nonlinear energy superposition is discussed
in Section <ref>.



§ TWO CLASSICAL FREE BOUNDARY PROBLEMS


A classical problem in fluid dynamics is the description
of a two-dimensional ideal fluid in terms of its stream
function, i.e. of a function whose level sets describe
the trajectories of the fluid.
For this, we consider an incompressible, irrotational
and inviscid fluid which occupies
a given planar region Ω⊂^2.
If V:Ω→^2 represents the velocity of the particles
of the fluid, the incompressibility condition implies that
the flow of the fluid through any portions of Ω
is zero (the amount of fluid coming in is exactly the
same as the one going out), that is, 
for any Ω_o⋐Ω, and denoting by ν
the exterior normal vector,

    0=∫_∂Ω_o∇ V·ν=
    ∫_Ω_o div V.

Since this is valid for any subdomain of Ω, we thus
infer that

    div V=0 Ω.

Now, we use that the fluid is irrotational to write equation (<ref>)
as a second order PDE. To this aim, let us analyze what a “vortex” is.
Roughly speaking, a vortex is given by a close trajectory,
say γ: S^1→Ω,
along which the fluid particles move. In this way, the velocity
field V is always parallel to the tangent direction γ'
and therefore

    0∫_S^1 V(γ(t))·γ'(t) dt
    =∮_γ V,

where the standard notation for the
circulation line integral is used.
That is, if we denote by S the region inside Ω
enclosed by the curve γ
(hence, γ=∂ S), 
we infer by (<ref>) and Stokes' Theorem that

    0∫_S  curl V· e_3,

where, as usual, we write {e_1,e_2,e_3}
to denote the standard basis of ^3,
we identify the vector V=(V_1,V_2)
with its three-dimensional image V=(V_1,V_2,0), and

    curl V(x):= ∇× V (x)=
    ( 
    
    e_1     e_2     e_3
    ∂_x_1   ∂_x_2   ∂_x_3
    
    V_1(x_1,x_2)   V_2(x_1,x_2)   0
    )=( ∂_x_1V_2(x)-∂_x_2V_1(x)) e_3.

In this setting, the fact that the fluid is irrotational is translated
in mathematical language into the fact that the opposite of (<ref>)
holds true, namely

    0=
    ∫_S  curl V· e_3,

for any S⊂Ω (say, with smooth boundary).
Since this is valid for any arbitrary region S, we thus
can translate the irrotational property of the fluid into 
the condition curl V=0 in Ω, that is

    ∂_x_1V_2-∂_x_2V_1=0 Ω.

Now, we consider the 1-form 

    ω:=
    V_2 dx_1-V_1 dx_2,
 
and we have that

    dω= -∂_x_2V_2 dx_1∧ dx_2-
    ∂_x_1V_1 dx_1∧ dx_2
    =- div V dx_1∧ dx_2
    =0,
 thanks to (<ref>).
Namely ω is closed, and thus exact (by 
Poincaré Lemma, at least if Ω is star-shaped).
This says that there exists a function u such that

    ω=du=∂_x_1u dx_1+
    ∂_x_2u dx_2.

By comparing this and (<ref>), we conclude that

    ∂_x_1u=V_2  ∂_x_2u=-V_1.

We observe that u is a stream function for the fluid, namely
the fluid particles move along the level sets of u:
indeed, if x(t) is the position of the fluid particle at time t,
we have that ẋ(t)=V(x(t)) is the velocity of the fluid, and

    d/dt u(x(t))=∂_x_1 u(x(t))ẋ_1(t)+
    ∂_x_2 u(x(t))ẋ_2(t)
            
    =
    ∂_x_1 u(x(t))V_1(x(t))+
    ∂_x_2 u(x(t))V_2(x(t))
    =V_2(x(t))V_1(x(t))-V_1(x(t))V_2(x(t))
    =0,

in view of (<ref>).

The stream function u also satisfies a natural overdetermined
problem. First of all, since ∂Ω represents the boundary
of the fluid, and the fluid motion occurs on the level sets of u,
up to constants we may assume that u=0 along ∂Ω.
In addition, along ∂Ω Bernoulli's Law prescribes
that the velocity is balanced by the pressure (which we take
here to be p=p(x)). That is,
up to dimensional constants, we can write that, along ∂Ω,

    p = |V|^2 =|∇ u|^2,

where we used again (<ref>) in the last identity.

Also, (<ref>) and (<ref>) give that, in Ω,

    Δ u=∂_x_1V_2-∂_x_2V_1=0,

that is, summarizing,

    {Δ u=0    Ω,
    
    u=0   ∂Ω,
    
    |∇ u|^2=p   ∂Ω.
    .

Notice that these types of overdetermined problems
are, in general, 
not solvable: namely, only “very special” domains
allow a solution of such overdetermined problem to exist
(see e.g. <cit.>). In this spirit, determining such
domain Ω is part of the problem itself, and
the boundary of Ω is, in this sense, a “free boundary”
to be determined together with the solution u.

These kinds of free boundary problems
have a natural formulation, which was widely studied
in <cit.>.
The idea is to consider an energy functional
which is the superposition of a Dirichlet part
and a volume term. By an appropriate domain variation,
one sees that minimizers (or, more generally, critical
points) of this functional correspond (at least in a weak sense)
to solutions of (<ref>) (compare, for instance,
the system in (<ref>)
here with Lemma 2.4 and Theorem 2.5 in <cit.>).
Needless to say, in this framework, the analysis of the minimizers
of this energy functional and of their level sets
becomes a crucial topic of research. 


In <cit.> a different energy functional
is taken into account, in which the volume term
is substituted by a perimeter term. This modification provides
a natural change in the free boundary condition (in this
setting, the pressure of the Bernoulli's Law is replaced
by the curvature of the level set, see formula (6.1)
in <cit.>).

In the following sections we will discuss what happens when:


  * we interpolate the volume term of
the energy functional of <cit.>
and the perimeter term of
the energy functional of <cit.>
with a fractional perimeter term, which recovers the volume
and the classical perimeter in the limit;

  * we consider a nonlinear energy superposition,
in which the total energy depends on the volume, or on the
(possibly fractional) perimeter, in a nonlinear fashion.




§ NONLOCAL FREE BOUNDARY PROBLEMS


A classical motivation for free boundary problems
comes from the superposition of
a “Dirichlet-type energy” D 
and an “interfacial energy” I.
Roughly speaking, one may consider the minimization problem
of an energy functional

    E:=D+I,

which takes into account the following two 
tendencies of the energy contributions, namely:


  * the term D
tries to reduce the oscillations of the minimizers,

  * while the term I penalizes the formation
of interfaces.

Two classical approaches
appear in the literature to measure these interfaces,
taking into account the “volume” of the phases
or the “perimeter” of the phase separations.
The first approach, based on a “bulk” energy contribution,
was widely studied in <cit.>.
In this setting, the energy superposition
in (<ref>) (with respect to
a reference domain Ω⊂^n) takes the form

    D=D(u):=∫_Ω |∇ u(x)|^2 dx
    
         I=I(u):=
    ∫_Ωχ_{ u>0}(x) dx
    = ℒ^n ( Ω∩{ u>0}),

where ℒ^n denotes, as customary,
the n-dimensional Lebesgue measure.
The case of two phase contributions
(namely, the one which takes into
account
the bulk energy of both {u>0} and {u<0})
was also considered in <cit.>.


The second approach, based on a “surface tension”
energy contribution,
was introduced in <cit.>.
In this setting, the energy superposition
in (<ref>) takes the form

    D=D(u):=∫_Ω |∇ u(x)|^2 dx
    
         I=I(u):=
     Per({ u>0} ,Ω),

where the notation

    Per(E,Ω):=∫_Ω |Dχ_E(x)| dx
    = [ χ_E]_BV(Ω)

represents the perimeter of the set E in Ω;
hence, if E has smooth boundary, then
Per(E,Ω)=ℋ^n-1((∂ E)∩Ω),
being ℋ^n-1 the (n-1)-dimensional
Hausdorff measure.

As pointed out in <cit.>,
the two free boundary problems
in (<ref>) and (<ref>)
can be settled into a unified framework,
and in fact they may be seen as “extremal” problems
of a family of energy functionals
indexed by a continuous parameter σ∈(0,1).

To this aim, given two measurable
sets E, F⊂^n, with ℒ^n(E∩ F)=0,
one considers the σ-interaction of E and F,
as given by the double integral

    𝒮_σ(E,F):=
    σ (1-σ) ∬_E× Fdx dy/|x-y|^n+σ.

In <cit.>, the notion
of σ-minimal surfaces has been introduced
by considering minimizers of the σ-perimeter
induced  by such interaction. Namely, one defines
the σ-perimeter of E in Ω
as the contribution relative to Ω
of the σ-interaction of E and its complement (which
we denote by E^c:=^n∖ E), that is

    Per_σ(E,Ω)
    := 𝒮_σ(E,E^c∩Ω)+
    𝒮_σ(E∩Ω,E^c∩Ω^c).

After <cit.>,
an intense activity has been performed to investigate
the regularity and the geometric properties
of σ-minimal surfaces:
see in particular <cit.>
for interior regularity results, and <cit.>
for the rather special behavior of
σ-minimal surfaces near the boundary
of the domain. See also <cit.>
for a recent survey on σ-minimal surfaces.


The analysis of the asymptotics of the σ-perimeter
as σ↗1 has been studied
under several perspectives in <cit.>. Roughly speaking,
up to normalization constants, we may say that Per_σ
approaches the classical perimeter as σ↗1.
On the other hand, as σ↘0,
we have that Per_σ
approaches the Lebesgue measure (again, up
to normalization constants,
see <cit.>
for precise statements and examples).

In virtue of these considerations,
we have that the free boundary problem
introduced in <cit.>,
which takes into account
the energy superposition
in (<ref>) of the form

    D=D(u):=∫_Ω |∇ u(x)|^2 dx
    
         I=I(u):=
     Per_σ({ u>0} ,Ω),

may be seen as an interpolation
of the problems stated in (<ref>) and (<ref>)
(that is, at least at a formal level, 
the energy functional in (<ref>) reduces to that
in (<ref>) as σ↘0
and to that in (<ref>)
as σ↗1). 

A nonlocal variation of the classical Dirichlet energy
has been also considered in <cit.>.
In this setting, the classical H^1-seminorm
in Ω of a function u is replaced
by a Gagliardo H^s-seminorm of the form

    s (1-s) ∬_Q_Ω|u(x)-u(y)|^2/|x-y|^n+2s dx dy,

where

    Q_Ω:=(Ω×Ω)∪
    (Ω×Ω^c)∪ (Ω^c×Ω)

and s∈(0,1). More precisely,
in <cit.> a superposition
of the Gagliardo seminorm
and the Lebesgue measure of the positivity
set is taken into account.

It is worth to point out that the domain Q_Ω
in (<ref>) comprises all the interactions
of points (x,y)∈^2n which involve the domain Ω,
since Q_Ω=(^n×^n)∖ 
(Ω^c×Ω^c).

In this sense, the integration over Q_Ω⊂^2n
is the natural counterpart of the classical integration over Ω
of the standard Dirichlet energy, since Ω=^n∖Ω^c.

Also, the double integral in (<ref>)
recovers the classical Dirichlet energy, see e.g. <cit.>.


In this spirit, in <cit.>
a fully nonlocal counterpart of the free boundary
problems in (<ref>) and (<ref>)
has been introduced, by studying
energy superpositions of Gagliardo norms
and fractional perimeters (see also <cit.>
for the 
superpositions of Gagliardo norms
and classical perimeters).
More precisely, 
the energy superposition
in (<ref>)
considered in <cit.> takes the form

    D=D(u):=
    s (1-s) ∬_Q_Ω|u(x)-u(y)|^2/|x-y|^n+2s dx dy
    
         I=I(u):=
     Per_σ({ u>0} ,Ω),

where s, σ∈(0,1).

We summarize here a series of results recently obtained
in <cit.>
for these nonlocal free boundary problems
(some of these results also rely on a notion of fractional harmonic
replacement analyzed in <cit.>).
First of all, we have that
minimizers[Here,
for simplicity,
we omit the fact that, in this setting, the minimization
is performed not only on a function, but
on a couple given by the function and its
positivity set. See Section 2 in <cit.>
for a rigorous discussion on this important
detail.]
of free boundary problems
with fractional perimeter interfaces
are continuous, possess suitable density estimates
and have smooth free boundaries
up to sets
of codimension 3:


[Theorems 1.1 and 1.2 in <cit.>]
Let u_⋆ be a minimizer of

    E(u):=∫_B_1 |∇ u(x)|^2 dx+
     Per_σ( {u>0},B_1),

with σ∈(0,1) and 0∈∂{u_⋆>0}.

Then u_⋆ is locally C^1- σ/2
and, for any r∈(0,1/2),

    ￼ min{ℒ^n(B_r∩{u_⋆≥0}), ℒ^n(B_r∩{u_⋆≤0}) }≥ cr^n,

for some c>0.

Moreover, the free boundary is a C^∞-hypersurface
possibly outside a small singular
set of Haussdorff dimension n-3.


We remark that the Hölder exponent 1- σ/2
is consistent with the natural scaling of the problem (namely u_r
(x) := r^σ/2-1 u_⋆ (rx) is still a minimizer).
Such type of regularity approaches the optimal exponent
in <cit.>
as σ↘0. Nevertheless, as σ↗1,
minimizers in <cit.> are known to be Lipschitz
continuous, therefore we think that it is a very interesting
open problem to investigate the optimal
regularity of u_⋆ in Theorem <ref>
(we stress that this optimal regularity may well approach
the Lipschitz regularity and so “beat the natural scaling
of the problem”).

Also, we think it is very interesting to obtain
optimal bounds on the dimension of the singular set
in Theorem <ref>.

It is also worth to observe that the minimizers
in Theorem <ref> satisfy a nonlocal free boundary
condition. Namely,
the normal jump J_⋆:=
|∇ u_⋆^+|^2-|∇ u_⋆^-|^2
along the smooth points
of the free boundary coincides (up to normalizing constants)
with the nonlocal mean curvature of the free boundary,
which is defined by

    𝒦^σ(x):=∫_^nχ_{ u≤0}(y)-χ_{ u>0}(y)/|x-y|^n+σ dy,

for x∈∂{ u>0}.

This free boundary condition has been presented in formula (1.6)
of <cit.>. Since 𝒦^σ approaches
the classical mean curvature as σ↗1
and a constant as σ↘0 (see e.g. <cit.>
and
Appendix B 
of <cit.>), we remark
that this nonlocal free boundary condition
recovers the classical ones in <cit.>
and in <cit.> as σ↗1,
and as σ↘0, respectively.


In <cit.>,
we consider the fully nonlocal case in which both the
energy components
become of nonlocal type,
namely we replace (<ref>) with the energy
functional

    E(u):=
    s (1-s) ∬_Q_B_1|u(x)-u(y)|^2/|x-y|^n+2s dx dy
    +
     Per_σ( {u>0},B_1),

with s, σ∈(0,1), where the notation in (<ref>)
has been also used.

In this setting, we have:

[Theorem 1.1 in <cit.>]
Let u_⋆ be a minimizer of (<ref>) 
with 0∈∂{u_⋆>0}.

Assume that u_⋆≥0 in B_1^c and that

    ∫_^n|u_⋆(x)|/1+|x|^n+2s dx<+∞.

Then, u_⋆ is locally C^s- σ/2
and, for any r∈(0,1/2),

    ￼ min{ℒ^n(B_r∩{u_⋆≥0}), ℒ^n(B_r∩{u_⋆=0}) }≥ cr^n,

for some c>0.


We observe that the Hölder exponent in Theorem <ref>
recovers that of Theorem <ref> as s↗1.
Once again, we think that it would be very interesting
to investigate the optimal regularity of the minimizers
in Theorem <ref>. Also,
Theorem <ref> has been established in the
“one-phase” case, i.e. under the assumption that the
minimizer has a sign. It would be very interesting
to establish similar results
in the “two-phase” case in which minimizers can change
sign. It is worth to remark that the case in which
minimizers change sign is conceptually
harder in the nonlocal setting
than in the local one, since the two phases interact between
each other, thus producing additional energy contributions
which need to be carefully taken into account.



§ NONLINEAR FREE BOUNDARY PROBLEMS


In <cit.>
a new class of free boundary problems
has been considered, by taking into account
“nonlinear energy superpositions”.
Namely, differently than in (<ref>),
the total energy functional considered in <cit.>
is of the form

    E:=D+Φ_0(I),

for a suitable function Φ_0.
When Φ_0 is linear, the energy functional in (<ref>)
boils down to its “linear counterpart”
given in (<ref>), but for a
nonlinear function Φ_0
the minimizers of the energy functional in (<ref>)
may exhibit[Let us give a brief motivation
for the nonlinear interface case. The classical energy functionals
in <cit.>
may be also considered in view of models arising in population 
dynamics.
Namely, one can consider the regions {u>0} and {u≤0}
as areas occupied by two different populations, that have
reciprocal “hostile” feelings. Then, the diffusive behavior
of the populations (which is
encoded by the Dirichlet term of the energy)
is influenced by the fact that the two populations will have the
tendency to minimize the contact between themselves,
and so to reduce an interfacial energy as much as possible.

In this setting, it is natural to consider the case
in which the reaction of the populations to the mutual contact
occurs in a nonlinear way. For instance, the case in which
additional irrationally motivated hostile feelings arise
from further contacts between the populations is naturally
modeled by a superlinear interfacial energy, while the
case in which the interactions between the populations
favor the possibility of compromises and cultural exchanges
is naturally
modeled by a sublinear interfacial energy.]
interesting differences with respect to the classical
case.

A detailed analysis of free boundary problems as in (<ref>)
is given in <cit.>.
Here, we summarize some of the results obtained
(we give here simpler statements, referring
to <cit.>
for more general results).
We take Φ_0:[0,+∞)→[0,+∞) to
be monotone, nondecreasing,
lower semicontinuous and coercive – in the sense that

    lim_t↗+∞Φ_0(t)=+∞.

We will also use the notation of writing Per_σ
for every σ∈[0,1], with the convention that


  * when σ∈(0,1), Per_σ
is the nonlocal perimeter defined in (<ref>);

  * when σ=1, Per_σ is the classical
perimeter;

  * when σ=0, Per_σ(E;Ω)=
ℒ^n(E∩Ω).

Then, in the spirit of (<ref>), we consider energy functionals of the form

    E(u):=∫_Ω |∇ u(x)|^2 dx+Φ_0(
     Per_σ( {u>0},Ω)).

Notice that, for σ∈(0,1) and Φ_0(t)=t,
the energy in (<ref>) reduces to that in (<ref>).
Similarly, for σ=0 and σ=1,
the energy in (<ref>)
boils down 
to those in <cit.> and <cit.>, respectively.

When σ=0,
a particularly interesting case of nonlinearity is given by Φ_0(t)=
t^n-1/n. Indeed, in this case, the interfacial energy
depends on the n-dimensional Lebesgue measure,
but it scales like an (n-1)-dimensional surface measure
(also, by Isoperimetric Inequality, the energy levels
of the functional in <cit.>
are above those in (<ref>)).

We point out that the free boundary problems in (<ref>)
develop a sort of natural instability, in the sense that
minimizers in a large ball, when restricted to smaller balls,
may lose their minimizing properties. In fact, minimizers
in large and small balls may be rather different from each
other:

[Theorem 1.1 
in <cit.>]
There exist
a nonlinearity Φ_0
and radii R_0>r_0>0 such that
a minimizer
for (<ref>) in Ω:=B_R_0
is not a minimizer
for (<ref>) in Ω:=B_r_0.


The counterexample in Theorem <ref>, which clearly shows the
lack[Just to recall the importance of scaling invariances in the classical free
boundary problems, let us quote page 114 of <cit.>:
“for (small) balls B_r [...] let us assume (see 3.1) B_r=B_1(0)”.]
of scaling invariance of the problem,
is constructed by taking advantage of the different
rates of scaling produced by a suitable nonlinear function Φ_0,
chosen to be constant on an interval.
Namely, the “saddle function” in the plane
u_0(x_1,x_2)=x_1x_2 is harmonic and therefore
minimizes the Dirichlet energy of (<ref>).
In large balls, the interface of u_0 (as well as the ones
of its competitors) produces a contribution that
lies in the constant part of Φ_0, thus reducing
the minimization problem of (<ref>) to the one
coming from the Dirichlet contribution, and so favoring u_0
itself. Viceversa, in small balls, the interface
of u_0 produces more (possibly fractional) perimeter than
the one of the competitors whose positivity sets do not
come to the origin, and this fact implies that u_0
is not a minimizer in small balls.



















This argument, which is depicted in Figures <ref> (A) and (B),
is rigorously explained in Section 3
of <cit.>.


In spite of this instability and of the lack of self-similar properties
of the energy functional, some regularity results
for minimizers of (<ref>) still hold true, under appropriate
assumptions on the nonlinearity Φ_0
(notice that, for Φ_0 with a constant part,
Theorem <ref> would produce the minimizer u_0
whose free boundary is a singular
cone, see Figure <ref> (A), hence any regularity
result on the free boundary has to rule 
out this possibility by a suitable assumption on Φ_0).
In this sense, we have the following results:


[Corollary 1.4 and Theorems 1.5 and 1.6
in <cit.>]
Let σ∈(0,1], Φ_0 be Lipschitz continuous
and strictly increasing.
Let u_⋆ be a minimizer of (<ref>)
in Ω:=B_R, with 0≤ u_⋆≤ M
on ∂ B_R, for some M>0.

Then, u_⋆∈ C^1-σ/2(B_R/4).
Also, for any r∈(0,R/4),

    ￼ min{ℒ^n(B_r∩{u_⋆≥0}), ℒ^n(B_r∩{u_⋆=0}) }≥ cr^n,

for some c>0.


For σ=0, a result similar to that in Theorem <ref>
holds true, in
the sense that u_⋆ is Lipschitz,
see Theorems 1.3, 8.1 and 9.2 in <cit.>.
Moreover, in this case one obtains additional results, such as the
nondegeneracy of the minimizers,
the partial regularity of the free boundary and the
full regularity in the plane:


[Theorems 1.4, 1.6 and 1.7
in <cit.>]
Let σ=0, Φ_0 be Lipschitz continuous
and strictly increasing.
Let u_⋆ be a minimizer of (<ref>)
in Ω, with 0∈∂{u_⋆>0}
(in the measure theoretic sense). 

Then, for any D⋐Ω,
there exists c>0 such that for any r>0 for which B_r⋐ D,
it holds that

    ∫_B_r∩{u_⋆>0} u_⋆^2(x) dx≥ c r^n+2.

Also, ∇ u_⋆ is locally BMO, in the sense that

    sup_B_r⋐ D_B_r| ∇ u_⋆(x)-
    ⟨∇ u_⋆⟩_r | dx
    ≤ C,

for some C>0, where

    ⟨∇ u_⋆⟩_r:=_B_r∇ u_⋆ (x) dx.

In addition ℋ^n-1(B_r∩
(∂{u_⋆>0})) <+∞.

Finally, if n=2, then B_r∩
(∂{u_⋆>0}) is made of continuously differentiable curves.


The BMO-type regularity and the partial regularity
of the free boundary in Theorem <ref>
rely in turn on some techniques developed in <cit.>.

It is also interesting to remark that the case σ=0
recovers the classical problems in <cit.>
after a blow-up:


[Theorem 1.5 and Proposition 10.1
in <cit.>]
Let σ=0, Φ_0 be Lipschitz continuous
and strictly increasing.
Let u_⋆ be a minimizer of (<ref>)
in Ω, with 0∈Ω. 

For any r>0, let u_r(x):=
 u_⋆ (rx)/r.
Then, there exists the blow-up limit

    u_0(x):=lim_r↘0 u_r(x).

Also, u_0 is continuous and with linear growth,
and it is a minimizer of the functional

    E_0(u):=
    ∫_B_ρ |∇ u(x)|^2 dx+
    λ_0 ℒ^n(B_ρ∩{u>0}),

where

    λ_0:= Φ_0'(
    ℒ^n(Ω∩{u_⋆>0})
    ).



We stress that the energy functional in (<ref>)
coincides with that analyzed in the classical paper <cit.>.
Nevertheless, the “scaling constant” λ_0
in (<ref>) depends on the original minimizer u_⋆,
as prescribed by (<ref>) (only in the case of a
linear Φ_0, we have that λ_0
is a structural constant independent of u_⋆).

The fact that geometric and physical quantities
arising in this type of problem are not universal constants
but depend on the minimizer itself
is, in our opinion, an intriguing feature of this type of
problems. In this sense, we recall that in <cit.>
the free boundary condition coincides with the classical
Bernoulli's law, namely
the normal jump J_⋆:=
|∇ u_⋆^+|^2-|∇ u_⋆^-|^2
along the smooth points
of the free boundary is constant
(in <cit.>
it coincides with the mean curvature
of the free boundary).
Differently from the classical cases,
in our nonlinear setting, the free boundary
condition depends on the minimizer itself.
Indeed, 
in our case the normal jump J_⋆ coincides with

    𝒦^σ Φ_0'( 
     Per_σ({u_⋆>0},Ω)),

where 𝒦^σ is the 
nonlocal mean curvature of the free boundary, as defined
in (<ref>) 
(see formula (1.12) in <cit.>
and formulas (1.13) and (1.14) in <cit.>).

We point out that (<ref>)
recovers the classical cases in <cit.>
when σ∈{0,1} and Φ_0 is linear.
On the other hand, when Φ_0 is not linear,
the free boundary condition in (<ref>)
takes into account the global behavior of the free boundary
and the (possibly fractional) perimeter of the minimizer
in the whole of the domain. In this sense,
this type of condition is “self-driven”, since it is influenced
by the minimizer itself and not only by the environmental conditions
and the structural constants.



§ REGULARITY OF STATIONARY POINTS OF THE ALT-CAFFARELLI FUNCTIONAL

In this section we would like to discuss some recent results on the further connections 
between the Alt-Caffarelli problem and the minimal surfaces. 
More specifically, we consider the stationary points (in particular minimizers) of the functional 

    E_AC[u]=∫_Ω|∇ u|^2+λ^2χ_{u>0}

and the capillarity surfaces in the sphere of radius λ
(notice that the critical points of the functional in (<ref>)
are related to the system
in (<ref>), see Theorem 2.5 in <cit.> for details on this).

The starting point of our analysis is to study the classical capillary drop problem. 


 §.§ Capillary drop problem

We first revisit the sessile drop problem and its higher dimensional analogue. 


We consider the functional 

    J(E):=∫_Ω_0|Dχ_E|+g∫_Ω_0x_n+1χ_Edx-∫_∂Ω_0λ(x')χ_Edx'

where 
Ω_0:={x=(x', x_n+1), x_n+1>0}, x'=(x_1, …, x_n)∈^n, 
g>0 is a given constant, |λ(x')|<1, and
χ_E is the characteristic function of E∈𝒜, where

    𝒜:={E⊂Ω_0 s.t.  E  has finite perimeter and ℋ^n+1(E)=V}.
 Here, the parameter V>0 is the volume fraction
of the droplet.

For n=2, the functional in (<ref>) is related to
the sessile drop problem, i.e. the problem of a 
(three-dimensional)
capillarity drop occupying the set E and sitting in the
halfspace {x_n+1>0}.

We observe that the first term in J(E) is the
energy due to the surface tension, 
the second term is the gravitational energy and the 
last term is the wetting energy which produces a contact angle 
θ(x') such that cosθ(x')=λ(x')
(see Figure <ref>).

By a Taylor expansion, we see that  

    √(1+|∇ u|^2)=1+1/2|∇ u|^2+…

Hence, if ∂ E is the graph of a (smooth) function u≥0 (with small gradient),
we obtain the approximation

    ∫_Ω_0|Dχ_E|=
    ℋ^n-1( (∂ E)∩Ω_0)=
    ∫_(∂Ω_0)∩{u>0}√(1+|∇ u|^2)
              =
    ∫_∂Ω_0χ_{u>0}+1/2∫_∂Ω_0|∇ u|^2+…

In other words, the functional E_AC 
in (<ref>)
is the linearization of the sessile drop problem
described by the functional J in (<ref>), with no gravity and constant
wetting energy density. This suggests
that there must be a strong link between the regularity of the minimizers of E_AC and the minimal surfaces.
We will now discuss[For completeness,
we recall that a nonlocal capillarity theory has been
recently developed in <cit.>.] in which sense this link rigorously occurs.



 §.§ Homogeneity of blow-ups and the support function

The first of such direct links was established in <cit.>, where it
is showed that the singular axisymmetric critical point of E_AC is an energy minimizer in dimension 7. This
singular energy minimizer
of the Alt-Caffarelli problem can be seen as the analog of the Simons cone

    S={x∈^8 : ∑_i=1^4x_i^2=∑_i=5^8x_i^2 },

which is an example of a singular hypersurface of least perimeter in dimension 8. The minimality of the Simons cone was 
first proved by E. Bombieri, E. De Giorgi and E. Giusti in <cit.>.

The cones with non-negative mean curvature arise naturally in the blow-up procedure 
of the minimizer u at a free boundary point. By Weiss' monotonicity formula (see <cit.>), 
any blow-up
limit u_0 of an energy minimizer of E_AC
is defined on ^n and
must be a homogeneous function of degree one.   

Let us write 

    u_0(x)=rg(σ)

where σ∈𝕊^n-1 (being 𝕊^n-1, as usual,
the unit sphere in ^n). Since u_0 is also a global minimizer of E_AC then it follows that Δ u_0=0 in Ω^+={u_0>0}∩^n.
Rewriting the equation Δ u_0=0 in polar coordinates we infer that g
is a solution of the equation 

    Δ_𝕊^n-1g+n g=0.

 Here Δ_𝕊^n-1g is the Laplace-Beltrami operator
 on the sphere. 
 We observe that u_0>0 if and only if g>0 and g=0 on the free boundary of u_0 which is a 
 cone due to the homogeneity of u_0. 
   
Equation (<ref>) can be rewritten as 

    Trace[g_ij+δ_ijg]=0.

It is well-known that 

    
 
determined by the parameterization 

    𝕊^n-1∋σ⟼ X(σ):=σ g(σ)+∇_σ g(σ),

see <cit.>.
In addition, we have that S and the sphere λ𝕊^n-1 are perpendicular at the 
contact points, see <cit.>.

In this sense, one can interpret g as the Minkowski support function of the surface S. In other words
X(σ)·σ =g(σ) and it is the distance of the tangent plane with normal σ from the origin.



 §.§ The mean radius equation


The previous discussion tells us that the sum of the principal radii of the surface S
is zero. Indeed, let κ_i=1/R_i, i=1, 2, …, n-1 be the ith principal curvature 
of S and R_i the corresponding principal radius. Then, in view of (<ref>),
it holds that
the matrix g_ij+δ_ijg
has eigenvalues 1/κ_1,…,1/κ_n-1,
and so its trace is equal to ∑_i=1^n-11/κ_i.
From this and (<ref>), we thus obtain that

    ∑_i=1^n-1R_i=∑_i=1^n-11/κ_i=0  in {g>0}.

This is called the mean radius equation. 
Recalling (<ref>),
the free boundary condition given in <cit.> (and corresponding
to the constancy of |∇ u_0|) along {g=0}
now becomes 
    |∇_σ g|=λ.
 This means that 
the surface S is contained in the sphere of radius λ. 

We point out that in dimension n=2 formula (<ref>) reduces to 

    κ_1+κ_2/κ_1κ_2=0

and therefore the mean curvature vanishes whenever the Gauss 
curvature is nonzero (i.e., κ_1κ_2≠0). 
If n≥ 3, then such simple interpretation is not possible.

In terms of the classification of global cones for the Alt-Caffarelli problem, we recall that
the
open question is whether for n=5, 6 the stable solution of the mean radius equation 
in λ𝕊^n-1 such that |∇_σ g|=λ on g=0 is
the disk passing through the origin (when n=2, this question is settled in <cit.>,
the case n=3 was addressed in <cit.> and n=4 was proved in <cit.>).

An open and challenging question is to classify the stationary solutions 
g and 
the corresponding zero mean radius surfaces of given topological type.



 §.§ Flame models

A closely related problem is the behavior of the solutions to 
the singular perturbation problem

    {[ Δ u_(x)=β_(u_)           B_1,;        |u_|≤ 1           B_1, ].

where >0 is small and

    {[        β_(t)=1/β(t/),;                      ; β(t)≥ 0,   β⊂ [0, 1],;                      ;      ∫_0^1β(t)dt= M>0 ].

is an approximation of the Dirac measure.

It is well known that (<ref>)
models propagation
of equidiffusional premixed  flames with high activation of energy, see <cit.>.

The limit u_0:=lim__j→ 0u__j (for a suitable sequence _j→ 0) solves a Bernoulli type free boundary problem
with the following free boundary condition

    | u^+|^2-| u^-|^2=2M.

In fact, it holds that u_0 is a stationary point of the Alt-Caffarelli functional
in (<ref>). 

If we choose {u_} to be a family of minimizers of the functional

    E_[u_]:=∫_Ω| u_|^2/2
    +(u_/),   (t)=∫_0^tβ(s)ds,

then u_ inherits the generic features of the
Alt-Caffarelli minimizers as described in <cit.>
(e.g. non-degeneracy, rectifiability of u, etc.).
Consequently, by
sending → 0, one can see that the limit u exists and it is a minimizer of
the Alt-Caffarelli functional 
    ∫_B_1| u|^2+2Mχ_{u>0}.

As it was mentioned above, the singular set of minimizers is empty in dimensions 2, 3 and  4. However, if u_ is not a minimizer then not much is known about the classification of the blow-ups of
the limit function u. An interesting question is to classify these stationary points of given topological type or Morse index. One recent result in this direction is that 
the associated surface S that we constructed via the support function is of ring type then 
S is a piece of catenoid, see <cit.>.

'
ADPM11
#1CODEN #1  #1ISBN #1   #1ISSN #1   #1LCCN #1   #1#1        URL           ath.sty               

[AC81]MR618549
H. W. Alt and L. A. Caffarelli.
Existence and regularity for a minimum problem with free boundary.
J. Reine Angew. Math., 325:0 105–144, 1981.
JRMAA8.
0075-4102.

[ACF84]MR732100
Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman.
Variational problems with two phases and their free boundaries.
Trans. Amer. Math. Soc., 2820 (2):0 431–461,
  1984.
TAMTAM.
0002-9947.
|http://dx.doi.org/10.2307/1999245|.

[ACKS01]MR1808651
I. Athanasopoulos, L. A. Caffarelli, C. Kenig, and S. Salsa.
An area-Dirichlet integral minimization problem.
Comm. Pure Appl. Math., 540 (4):0 479–499,
  2001.
CPAMA.
0010-3640.
|http://dx.doi.org/10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.3.CO;2-U|.

[ADPM11]MR2765717
Luigi Ambrosio, Guido De Philippis, and Luca Martinazzi.
Gamma-convergence of nonlocal perimeter functionals.
Manuscripta Math., 1340 (3-4):0 377–403, 2011.
MSMHB2.
0025-2611.
|http://dx.doi.org/10.1007/s00229-010-0399-4|.

[Ale39]Aleksandrov
A. Alexandroff.
über die Oberflächenfunktion eines konvexen Körpers.
  (Bemerkung zur Arbeit “Zur Theorie der gemischten Volumina von
  konvexen Körpern”).
Rec. Math. N.S. [Mat. Sbornik], 6(48):0 167–174, 1939.

[AV14]MR3230079
Nicola Abatangelo and Enrico Valdinoci.
A notion of nonlocal curvature.
Numer. Funct. Anal. Optim., 350 (7-9):0
  793–815, 2014.
0163-0563.
|http://dx.doi.org/10.1080/01630563.2014.901837|.

[BBM02]MR1945278
Jean Bourgain, Haïm Brezis, and Petru Mironescu.
Limiting embedding theorems for W^s,p when s↑1 and
  applications.
J. Anal. Math., 87:0 77–101, 2002.
JOAMAV.
0021-7670.
|http://dx.doi.org/10.1007/BF02868470|.
Dedicated to the memory of Thomas H. Wolff.

[BDGG69]MR0250205
E. Bombieri, E. De Giorgi, and E. Giusti.
Minimal cones and the Bernstein problem.
Invent. Math., 7:0 243–268, 1969.
0020-9910.
|http://dx.doi.org/10.1007/BF01404309|.

[BFV14]MR3331523
Begoña Barrios, Alessio Figalli, and Enrico Valdinoci.
Bootstrap regularity for integro-differential operators and its
  application to nonlocal minimal surfaces.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 130
  (3):0 609–639, 2014.
0391-173X.

[BLV16]2016arXiv161208295B
Claudia Bucur, Luca Lombardini, and Enrico Valdinoci.
Complete stickiness of nonlocal minimal surfaces for small values of
  the fractional parameter.
ArXiv e-prints, December 2016.

[BN16]MR3556344
Haïm Brezis and Hoai-Minh Nguyen.
The BBM formula revisited.
Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 270
  (4):0 515–533, 2016.
1120-6330.
|http://dx.doi.org/10.4171/RLM/746|.

[Caf95]MR1347971
Luis A. Caffarelli.
Uniform Lipschitz regularity of a singular perturbation problem.
Differential Integral Equations, 80 (7):0
  1585–1590, 1995.
0893-4983.

[CDSS16]MR3532394
L. Caffarelli, D. De Silva, and O. Savin.
Obstacle-type problems for minimal surfaces.
Comm. Partial Differential Equations, 410 (8):0
  1303–1323, 2016.
CPDIDZ.
0360-5302.
|http://dx.doi.org/10.1080/03605302.2016.1192646|.

[CJK04]CJK
Luis A. Caffarelli, David Jerison, and Carlos E. Kenig.
Global energy minimizers for free boundary problems and full
  regularity in three dimensions.
In Noncompact problems at the intersection of geometry,
  analysis, and topology, volume 350 of Contemp. Math., pages 83–97.
  Amer. Math. Soc., Providence, RI, 2004.
|http://dx.doi.org/10.1090/conm/350/06339|.

[CRS10a]MR2675483
L. Caffarelli, J.-M. Roquejoffre, and O. Savin.
Nonlocal minimal surfaces.
Comm. Pure Appl. Math., 630 (9):0 1111–1144,
  2010.
CPAMA.
0010-3640.
|http://dx.doi.org/10.1002/cpa.20331|.

[CRS10b]MR2677613
Luis A. Caffarelli, Jean-Michel Roquejoffre, and Yannick Sire.
Variational problems for free boundaries for the fractional
  Laplacian.
J. Eur. Math. Soc. (JEMS), 120 (5):0
  1151–1179, 2010.
1435-9855.
|http://dx.doi.org/10.4171/JEMS/226|.

[CSV15]MR3390089
Luis Caffarelli, Ovidiu Savin, and Enrico Valdinoci.
Minimization of a fractional perimeter-Dirichlet integral
  functional.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 320
  (4):0 901–924, 2015.
0294-1449.
|http://dx.doi.org/10.1016/j.anihpc.2014.04.004|.

[CV11]MR2782803
Luis Caffarelli and Enrico Valdinoci.
Uniform estimates and limiting arguments for nonlocal minimal
  surfaces.
Calc. Var. Partial Differential Equations, 410
  (1-2):0 203–240, 2011.
0944-2669.
|http://dx.doi.org/10.1007/s00526-010-0359-6|.

[CV13]MR3107529
Luis Caffarelli and Enrico Valdinoci.
Regularity properties of nonlocal minimal surfaces via limiting
  arguments.
Adv. Math., 248:0 843–871, 2013.
0001-8708.
|http://dx.doi.org/10.1016/j.aim.2013.08.007|.

[Dáv02]MR1942130
J. Dávila.
On an open question about functions of bounded variation.
Calc. Var. Partial Differential Equations, 150
  (4):0 519–527, 2002.
0944-2669.
|http://dx.doi.org/10.1007/s005260100135|.

[DFPV13]MR3007726
Serena Dipierro, Alessio Figalli, Giampiero Palatucci, and Enrico Valdinoci.
Asymptotics of the s-perimeter as s↘0.
Discrete Contin. Dyn. Syst., 330 (7):0
  2777–2790, 2013.
1078-0947.
|http://dx.doi.org/10.3934/dcds.2013.33.2777|.

[Dip14]MR3506705
S. Dipierro.
Asymptotics of fractional perimeter functionals and related problems.
Rend. Semin. Mat. Univ. Politec. Torino, 720
  (1-2):0 3–16, 2014.
0373-1243.

[DK15]2015arXiv150807447D
Serena Dipierro and Aram L. Karakhanyan.
Stratification of free boundary points for a two-phase variational
  problem.
ArXiv e-prints, August 2015.

[DKV15]2015arXiv151203043D
Serena Dipierro, Aram Karakhanyan, and Enrico Valdinoci.
A class of unstable free boundary problems.
ArXiv e-prints, December 2015.

[DKV16]2016arXiv161100412D
Serena Dipierro, Aram Karakhanyan, and Enrico Valdinoci.
A nonlinear free boundary problem with a self-driven Bernoulli
  condition.
ArXiv e-prints, November 2016.

[DLV]LUCA
Serena Dipierro, Luca Lombardini, and Enrico Valdinoci.
A free boundary problem: superposition of nonlocal energy plus
  classical perimeter.
Preprint.

[DMV16]2016arXiv161000075D
Serena Dipierro, Francesco Maggi, and Enrico Valdinoci.
Asymptotic expansions of the contact angle in nonlocal capillarity
  problems.
ArXiv e-prints, September 2016.

[DNPV12]MR2944369
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.
Hitchhiker's guide to the fractional Sobolev spaces.
Bull. Sci. Math., 1360 (5):0 521–573, 2012.
0007-4497.
|http://dx.doi.org/10.1016/j.bulsci.2011.12.004|.

[DSJ09]MR2572253
Daniela De Silva and David Jerison.
A singular energy minimizing free boundary.
J. Reine Angew. Math., 635:0 1–21, 2009.
0075-4102.
|http://dx.doi.org/10.1515/CRELLE.2009.074|.

[DSR12]MR2926238
D. De Silva and J. M. Roquejoffre.
Regularity in a one-phase free boundary problem for the fractional
  Laplacian.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 290
  (3):0 335–367, 2012.
0294-1449.
|http://dx.doi.org/10.1016/j.anihpc.2011.11.003|.

[DSV15]MR3427047
Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci.
A nonlocal free boundary problem.
SIAM J. Math. Anal., 470 (6):0 4559–4605,
  2015.
0036-1410.
|http://dx.doi.org/10.1137/140999712|.

[DSV16a]Dipierro2016JFA
Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci.
Boundary behavior of nonlocal minimal surfaces.
J. Funct. Anal., 2016.
0022-1236.
|http://www.sciencedirect.com/science/article/pii/S0022123616303858|.

[DSV16b]MR3516886
Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci.
Graph properties for nonlocal minimal surfaces.
Calc. Var. Partial Differential Equations, 550
  (4):0 Paper No. 86, 25, 2016.
0944-2669.
|http://dx.doi.org/10.1007/s00526-016-1020-9|.

[DV15]MR3320130
Serena Dipierro and Enrico Valdinoci.
On a fractional harmonic replacement.
Discrete Contin. Dyn. Syst., 350 (8):0
  3377–3392, 2015.
1078-0947.
|http://dx.doi.org/10.3934/dcds.2015.35.3377|.

[DV16]Dipierro2016HP
Serena Dipierro and Enrico Valdinoci.
Continuity and density results for a one-phase nonlocal free boundary
  problem.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 2016.
0294-1449.
|http://www.sciencedirect.com/science/article/pii/S0294144916300853|.

[DV17]2016arXiv160706872D
Serena Dipierro and Enrico Valdinoci.
Nonlocal minimal surfaces: interior regularity, quantitative
  estimates and boundary stickiness.
Recent Developments in the Nonlocal Theory (T. Kuusi, G.
  Palatucci eds.). Book Series on Measure Theory. De Gruyter, Berlin, 2017.

[FV15]FIGALLI
Alessio Figalli and Enrico Valdinoci.
Regularity and bernstein-type results for nonlocal minimal surfaces.
J. Reine Angew. Math., 2015.
0075-4102.
|https://www.degruyter.com/view/j/crll.ahead-of-print/crelle-2015-0006/crelle-2015-0006.xml|.

[JS15]JS
David Jerison and Ovidiu Savin.
Some remarks on stability of cones for the one-phase free boundary
  problem.
Geom. Funct. Anal., 250 (4):0 1240–1257, 2015.
1016-443X.
|http://dx.doi.org/10.1007/s00039-015-0335-6|.

[Kar16]AK
Aram Karakhanyan.
Minimal surfaces arising in singular perturbation problems.
Preprint, 2016.

[MS02]MR1940355
V. Mazya and T. Shaposhnikova.
On the Bourgain, Brezis, and Mironescu theorem concerning
  limiting embeddings of fractional Sobolev spaces.
J. Funct. Anal., 1950 (2):0 230–238, 2002.
JFUAAW.
0022-1236.
|http://dx.doi.org/10.1006/jfan.2002.3955|.

[MV16]2016arXiv160608610M
Francesco Maggi and Enrico Valdinoci.
Capillarity problems with nonlocal surface tension energies.
ArXiv e-prints, June 2016.

[Pon04]MR2033060
Augusto C. Ponce.
A new approach to Sobolev spaces and connections to
  Γ-convergence.
Calc. Var. Partial Differential Equations, 190
  (3):0 229–255, 2004.
0944-2669.
|http://dx.doi.org/10.1007/s00526-003-0195-z|.

[Ser71]MR0333220
James Serrin.
A symmetry problem in potential theory.
Arch. Rational Mech. Anal., 43:0 304–318, 1971.
0003-9527.

[SV13a]MR3090533
Ovidiu Savin and Enrico Valdinoci.
Regularity of nonlocal minimal cones in dimension 2.
Calc. Var. Partial Differential Equations, 480
  (1-2):0 33–39, 2013.
0944-2669.
|http://dx.doi.org/10.1007/s00526-012-0539-7|.

[SV13b]MR3035063
Ovidiu Savin and Enrico Valdinoci.
Some monotonicity results for minimizers in the calculus of
  variations.
J. Funct. Anal., 2640 (10):0 2469–2496, 2013.
JFUAAW.
0022-1236.
|http://dx.doi.org/10.1016/j.jfa.2013.02.005|.

[Wei98]weiss
Georg S. Weiss.
Partial regularity for weak solutions of an elliptic free boundary
  problem.
Comm. Partial Differential Equations, 230
  (3-4):0 439–455, 1998.
0360-5302.
|http://dx.doi.org/10.1080/03605309808821352|.






