Uniqueness and Stability of Optimizers for a Membrane Problem
Behrouz Emamizadeh Amin Farjudian Yichen Liu Monica Marras
math.AP math.OC 65K10, 74H55, 35J20, 35J25










Uniqueness and Stability of Optimizers for a Membrane
  Problem
    Behrouz EmamizadehSchool of Mathematical Sciences,
    University of Nottingham Ningbo China, 199 Taikang East Road,
    Ningbo, Zhejiang China 315100,
    mailto:Behrouz.Emamizadeh@nottingham.edu.cnBehrouz.Emamizadeh@nottingham.edu.cn
  Amin FarjudianSchool of Computer Science, University of
    Nottingham Ningbo China, 199 Taikang East Road, Ningbo, Zhejiang
    China 315100,
    mailto:Amin.Farjudian@gmail.comAmin.Farjudian@gmail.com 
  Corresponding author Yichen LiuBeijing
    International Center for Mathematical Research, Peking University,
    China,
    mailto:yichen.liu07@yahoo.comyichen.liu07@yahoo.com Monica MarrasDipartimento di Matematica e Informatica,
    Università di Cagliari, viale Merello 92, 09123 Cagliari, Italy,
    mailto:mmarras@unica.itmmarras@unica.it
    
========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





  We investigate a PDE-constrained optimization problem, with an
  intuitive interpretation in terms of the design of robust membranes
  made out of an arbitrary number of different materials. We prove
  existence and uniqueness of solutions for general smooth bounded
  domains, and derive a symmetry result for radial ones. We strengthen
  our analysis by proving that, for this particular problem, there are
  no non-global local optima. When the membrane is made out of two
  materials, the problem reduces to a shape optimization problem. We
  lay the preliminary foundation for computable analysis of this type
  of problem by proving stability of solutions with respect to some of
  the parameters involved.  






  Key Words: Optimization, Stability, Radial symmetry, Boundary
  value problem, Rearrangements of functions.

  Mathematics Subject Classification: 65K10, 74H55, 35J20, 35J25.









§ INTRODUCTION


Consider the boundary value
problem:


    {1ex1.2[ -Δ u+g(x)u=f(x),            in D,;             u=0,          on ∂ D, ].


in which D⊆^N is a smooth domain, N ∈{ 2,3 }, g
is a non-negative function in L^∞(D), and f is a non-negative
function in L^2(D).

When the range of the function g is a finite set, say, {α_1,
α_2, …, α_n}, this equation may be interpreted in a
very intuitive way. Indeed, the boundary value
problem (<ref>) models an elastic membrane, constructed
out of n different materials, fixed around the boundary, and subject
to a vertical force f(x) at each point x. The solution u denotes
the displacement of the membrane from the rest position.


Let us assume that we have been given the n constituent materials,
together with the force f and the geometry of the domain, and our
task is to construct a robust membrane out of the given
materials. In this paper, we demonstrate how this may be
achieved. More formally, we associate the following energy
functional with the boundary value problem (<ref>):


    Φ(g) ∫_Dfu_g  x=∫_D|∇ u_g|^2  x+∫_Dgu_g^2  x,


in which, u_g is the unique solution of (<ref>). At an
intuitive level, this energy functional is mean to measure the
vulnerability of the membrane. It should be straightforward to
verify that the following identity follows from the variational
formulation of u_g:


    Φ(g)=sup_v∈ H^1_0(D){2∫_Dfv  x-∫_D(|∇
        v|^2+gv^2)  x}.


We assume that the information about the constituent materials is
provided in a given function g_0 which satisfies 0≤ g_0≤ 1,
and which is is not identically zero. We let
R≡R(g_0) denote the rearrangement class
generated by g_0 (Definition <ref>
[below]eq:rearrangeclass). To obtain a robust membrane, we
need to obtain the arrangement of g_0 with the least vulnerability,
, we need to solve the following minimization problem:


    inf_g∈RΦ(g).




  The minimization problem (<ref>) is of interest from
  a pure mathematical perspective as well. Indeed, the maximum
  principle ensures that u_g, the solution of (<ref>),
  is positive. Hence, the integral ∫_D fu_g   x is the
  L^1(μ)-norm of u_g, with dμ being the measure which is
  absolutely continuous with respect to the Lebesgue measure x,
  having f as its Radon-Nikodym derivative with respect to
  x. Minimization of various norms of solutions of partial
  differential equations is a classical topic of interest among
  mathematicians.  






 §.§ Approach and contributions


Our approach towards proving the solvability of
(<ref>) is based on the well-developed theory of
rearrangements of
  functions <cit.>. Specifically,
we use the theory developed by
G. R. Burton <cit.> for optimization over
rearrangement classes. To this end, we first relax the minimization
problem (<ref>) by extending the admissible set
R to its weak closure R with respect to
L^2-topology. Once the relaxed problem is shown to be solvable, we
will demonstrate how the appropriate restrictions on the force
function f imply that solutions of the relaxed problem are indeed
solutions of the original problem (<ref>).


We strengthen our results by proving that the optimization
problem (<ref>) has no non-global local optima, and by
showing that, when D is a ball and f is radial, then the solution
of (<ref>) is radial and non-increasing. 

 An appealing aspect of our method is that it can also
  be used when the function g belongs to the larger class L^p(D),
  for 1<p<∞, in which case, only minor modifications will be
  required. We prefer, however, to focus on the case g∈ L^∞
  (D), in order to minimize technicalities, and keep the model more
  realistic.



In the second part of the paper, we discuss some stability
results. These results are of utmost importance in setting up a
framework for computable analysis of problems such as our main
problem (<ref>). 





 §.§ Related work


For any given set E ⊆ D, by χ_E we denote the
characteristic function of E, , χ_E(x)=1 if x∈ E, and
χ_E(x)=0 if x ∉ E. Henrot and
Maillot <cit.> have
investigated the special case of the minimization problem
(<ref>), in which g_0=χ_E_0, for some E_0
⊆ D with |E_0|=α. Under this assumption, one would get
R={χ_E: E ⊆ D ∧ |E|=α}. In simple
terms, the rearrangement class generated by g_0 would be exactly the
set of all characteristic functions of those measurable subsets of D
that have the same Lebesgue measure as E_0.

Henrot and Maillot <cit.>
prove the solvability for this special case, and state the minimality
condition in terms of tangent cones. Since the underlying function
space is L^∞(D), they are able to derive a convenient
formulation of the tangent cone of an appropriate convex set.

The method employed in <cit.>
is inadequate for addressing the optimization problem
(<ref>) for general generators g_0. The theory that
we shall introduce in this paper, however, not only furnishes an
answer to the aforementioned question, but also can be used for a
broader range that includes other design problems.

The second part of the current paper addresses some further issues,
including stability properties of the solutions. This is part of a
broader programme of laying the foundations for robust computable
analysis of rearrangement optimization problems in particular, and
shape optimization problems in general. In this regard, we have
carried out some general stability analyses pertaining to
rearrangement optimization classes, which may be found
in <cit.>.



  Parts of an earlier draft of this article have appeared in the PhD
  dissertation of one of the
  co-authors <cit.>.






 §.§ Structure of the paper


The remainder of the paper is structured as follows:



  * Section <ref> contains preliminary material
  from the theory of rearrangements of functions.


  * In Section <ref> we prove existence and
  uniqueness of optimal solutions, and provide a radial symmetry
  result as well. For the minimization problem, we will show that
  there are no non-global local optima. Finally, we provide some
  remarks on the corresponding maximization problem.


  * In Section <ref>, we discuss the shape
  optimization variant of the main problem. Specifically, we will
  discuss monotonicity and stability results related to the case where
  the generator is two-valued.


  * In Section <ref>, we provide some
  remarks on the numerical simulation of the optimization problem.

  * In order to avoid breaking the flow of the paper, the lengthy
  proof of Lemma <ref> (from
  Section <ref>) is moved to
  Section <ref>.


  * In Section <ref>, we finish the paper
  with some concluding remarks.








§ PRELIMINARIES



In this section, we recall some well-known results from the theory of
rearrangements of functions. Henceforth, we denote the N-dimensional
Lebesgue measure of a measurable set E by |E|. Moreover, for a
Lebesgue measurable function h:D→[0,∞) and α≥ 0,
we let:


    λ_h(α) | {
      x∈ D: h(x)≥α}|.





  
  Let g, g_0:D→[0,∞) be Lebesgue measurable. We say that g is a
  rearrangement of g_0 if and only if ∀α≥ 0: λ_g_0(α) = λ_g(α).



   For a Lebesgue measurable
  g:D→[0,∞), the essentially unique decreasing rearrangement
  g^Δ is defined on (0,|D|) by g^Δ(s) max{α:λ_g(α)≥ s}. The essentially
  unique increasing rearrangement g_Δ of g is defined by
  g_Δ(s)  g^Δ(|D|-s).




The set R≡R(g_0), called the rearrangement class generated by g_0, is defined as follows


    R(g_0) {g:D→[0,∞):g  is a rearrangement of  g_0}.





  For a function f:D→[0,∞), we say that the graph of f has
  no significant flat sections on D if ∀ c ≥ 0: |{x∈
      D:f(x)=c}|=0.


Henceforth, the support of g will be denoted by
S(g)≡{x∈ D:g(x)>0}, and the reader should
distinguish this definition of support from the usual topological
definition. We use R to denote the weak closure of
R in L^2(D). It is well-known that R is
convex, and weakly compact in L^2(D).


   Let R be the weak closure
  of R in L^2(D). Then, R⊆
  L^∞(D), and ∀ g∈R:
  g_∞≤g_0_∞.



  In order to derive a contradiction, we suppose g∉
  L^∞(D). Hence, for every positive M:

  
    |{x∈
          D:g(x)>M}|>0.


  Let us choose M=g_0_∞, and set E {x∈
    D:g(x)>g_0_∞}. Since
  g∈R, there exists
  {g_n}⊆R such that g_n⇀
  g in L^2(D). Then, we have:


    ∫_E g_n  x=∫_D g_nχ_E x→∫_D gχ_E x=∫_E g  x.


From the definition of E and the fact that ∫_Eg_n  x≤g_0_∞|E|, in conjunction with (<ref>), we
deduce:


    g_0_∞|E|<∫_Eg  x=lim_n→∞∫_E
    g_n  x≤g_0_∞|E|.


Obviously, (<ref>) is a contradiction. The above argument implies
that the measure of E is zero.  Hence,
g_∞≤g_0_∞. This completes
the proof of the lemma.



  
  Suppose {g_n}⊆ L_+^∞(D), and g∈
  L^2(D). Suppose g_n⇀ g in L^2(D).  Then, g is
  non-negative a.e. in D.


 This is an immediate consequence of Mazur's Lemma. Indeed,
  by Mazur's Lemma, there exists a sequence {v_n} in the
  convex hull of the set {g_n:n∈ℕ} such that
  v_n→ g in L^2(D). Therefore, v_n→ g in measure. Whence,
  there exists a subsequence of {v_n} which converges
  to g a.e. in D. This completes the proof.


The next lemma is easy to prove:



Suppose that f:D→ [0,∞) is measurable. Then, for every
measurable subset E⊆ D:

  
    ∫_Ef  x≥∫_0^|E|f_Δ(s)  s.






For every g in R we have |S(g_0)|≤|S(g)|.



  In order to derive a contradiction, let us assume that
  |S(g)|<|S(g_0)|. Hence,
  α≡∫_0^|S(g)^c|g_0_Δ x is positive.  Since
  g∈R, there exists
  {g_n}⊆R such that g_n⇀
  g in L^2(D). Then, we have:

  
    α=∫_0^|S(g)^c|g_0_Δ x=∫_0^|S(g)^c|g_n_Δ x ≤∫_S(g)^c g_n  x
         =
        ∫_D g_nχ_S(g)^c x→∫_D gχ_S(g)^c x=∫_S(g)^c g  x=0,


  which is a contradiction. The inequality in (<ref>) is a
  consequence of Lemma <ref>.



We make use of the following lemmata from <cit.> and <cit.>.


  
  The following characterization for the weak closure of R
  holds:
  
    R
     =
     {g∈ L^1(D): ∫_Dg  x = ∫_Dg_0  x   and  ∀ s∈
       (0,|D|): ∫_0^sg^Δ t≤∫_0^sg_0^Δ t
     } .



 See Lemma 2.3 in <cit.>.


In line with the established convention
of <cit.>, in what follows we often write
`increasing' instead of non-decreasing and `decreasing' instead of
non-increasing.


  
  Suppose that f:D→ [0,∞) is measurable and has no
  significant flat sections on D. Then, there exists an increasing
  function ψ such that ψ(f) is a rearrangement of g_0.
  Moreover, there is a decreasing function ψ̃ such that
  ψ̃(f) is a rearrangement of g_0.



  See Lemma 2.9 in <cit.>.




  
  Let f∈ L^2(D) be a non-negative and non-trivial function (,
  it is not identically zero), and assume that there is an increasing
  function ψ such that ψ(f)∈R. Then ψ(f) is
  the unique maximizer of the linear functional L(h) ∫_D fh   x relative to h∈R.



  See Lemma 2.4 in <cit.>.


We will also need the following rearrangement result for the Dirichlet
integral (see, , <cit.>). Note that
here v^* denotes the Schwarz symmetrization of v (see,
, <cit.>):


 
  
  
  
  *  If v∈
    H^1_0(^N) is non-negative, then, v^*∈ H^1_0(^N), and the
    following inequality holds:
    
    ∫_^N|∇ v^*|^2  x≤∫_^N|∇ v|^2  x.


  
  *  If v∈ H^1_0(^N) is
    non-negative and equality holds in (<ref>), then, for every
    0≤α<M:=ess sup v, v^-1(α,∞) is a
    translate of the disk v^*^-1(α,∞), almost
    everywhere. If, in addition, {x∈^N: ∇ v=0, 0<v(x)<M}
    has zero measure, then v is a translate of v^*.
  







§ OPTIMAL SOLUTIONS




We need to make certain assumptions on the force function f in order
to be able to obtain our main results. Henceforth, v_f∈ H_0^1(D)
will denote the unique solution of the Poisson boundary value problem:


    {0.3ex1.2[  -Δ v_f       =       f   in D,;     v_f       =       0 on ∂ D. ].


Here is the main assumption on which our results will hinge:



  [t]0.6
      

        A1:  v_f≤ f, in D.
      





A minor problem with this assumption is that its statement involves
the solution to the Poisson boundary value problem
(<ref>). It turns out that we can also work with the
following assumption, whose statement involves just the function f
and its Laplacian:



      

        A2:  f≤-Δ f, in D.
      






A2 implies A1.



  Notice that we have:


    {1ex1.2[ -Δ (v_f-f)=f+Δ f             in D;          v_f-f≤0    on ∂
          D. ].


Since f+Δ f is non-positive, we can apply the maximum principle
to (<ref>) to deduce v_f≤ f.


As a consequence, all of the results that will be proved based
  on A1 will also hold for assumption A2.



  Our assumptions are valid, in the sense that there are non-negative
  functions satisfying A2, and by implication A1. Indeed,
  consider the boundary value problem
  
    {0.3ex1.2[  -Δ u-u       =       N   in D,;       u       =       0 on ∂ D, ].


  in which N∈[0,∞). The energy functional associated with
  (<ref>) is:

  
    I(u)=1/2∫_D|∇
        u|^2  x-1/2∫_D u^2  x-∫_DNu  x.

 
  It is clear from the Poincaré inequality that, if
  D is thin, then I(u) will be coercive. So, by an application of
  the direct method of calculus of variations to the functional
  I(u), we infer the existence of a critical point which is a
  solution of (<ref>). In order to show that (<ref>) has a
  non-negative solution, it suffices to point out that I(|u|)≤
  I(u).





 §.§ Existence, uniqueness, and optimality condition



Our assumptions guarantee that the solution u_g of the boundary
value problems (<ref>) has no significant flat sections on
S(g), a fact which will be used in the proof of our main result:


  
  Suppose that f satisfies assumption A1, and g is a
  measurable function such that 0≤ g≤ 1. Then, u_g has no
  significant flat sections on S(g).



  From the boundary value problems (<ref>) and (<ref>), we
  deduce:
  
    {1ex1.2[ -Δ (u_g-v_f)+g(u_g-v_f)=-gv_f,                         in  D,;                     u_g-v_f=0,                        on ∂ D. ].


  Since g and v_f are non-negative, u_g<v_f in D by the strong
  maximum principle.

  In order to derive a contradiction, we assume that there exists an
  L⊆ S(g) such that the measure of L is positive, and
  u_g is constant on L. By applying Lemma 7.7
  in <cit.>, we
  infer f=gu_g in L. Hence:

    f=gu_g<gv_f≤ v_f ≤ f,  in  L,

which is a contradiction.


Next, we turn to the energy functional. In order to prove the
existence and uniqueness of solutions of the minimization problem
(<ref>), we need the following basic result regarding
the energy functional Φ:

   The energy functional Φ satisfies the
  following:
  
  
  *  Φ is weakly continuous on R with
    respect to the L^2-topology.
  
  *  Φ is strictly convex on R.
  
  *  Given g and h in
    R, the following formula holds:

    
    lim_t→
            0^+Φ(ξ_t)-Φ(g)/t=-∫_D(h-g)u^2  x,  
          0<t<1,


    in which ξ_t=g+t(h-g), and u=u_g.
  



  The proof of this lemma is quite long and involved. In order not to
  break the flow of the discussion, the proof is placed in a separate
  section altogether. Please see Sect. <ref>.




The main result of the paper is the following:


  
  Suppose that f satisfies assumption A1. Then the
  minimization problem (<ref>) has a unique solution
  ĝ∈ R. Moreover, there exists an increasing function
  ψ such that:
  
    ĝ=ψ(û)  a.e. in  D,

  where û=u_ĝ.


 

  We relax the minimization problem (<ref>) first by
  extending the admissible set R to
  R. Thus, we consider:

  
    inf_g∈RΦ(g)


  By Lemma <ref> (<ref>), Φ is
  weakly continuous on R with respect to the
  L^2-topology. Hence, the minimization problem (<ref>)
  is solvable. Furthermore, thanks to the strict convexity of Φ
  (Lemma <ref> (<ref>)) the solution to
  (<ref>) is unique.  Let us denote this solution by
  ĝ.

  Fix g∈R, and set g_t=ĝ+t(g-ĝ),
  for t∈ (0,1). Due to the convexity of R,
  g_t∈R. From Lemma
  <ref> (<ref>) we can derive
  ∫_D(g-ĝ)û^2  x≤0. Whence, ĝ maximizes the
  linear functional L(h) ∫_Dhû^2  x, relative to
  h∈R. From Lemma <ref> and Lemma
  <ref>, it follows that the graph of û_S, the
  restriction of û to the set S(ĝ), has no significant
  flat sections on S(ĝ). From Lemma <ref>, we know
  that there exists a g_1∈R such that S(g_1)⊆
  S(ĝ). Therefore, if we denote by R_S the functions
  which are rearrangements of g_1 on S(ĝ), then by Lemma
  <ref> we infer the existence of an increasing function
  ψ_S such that ψ_S((û)^2_S)∈ R_S. We now
  proceed to extending ψ_S to an increasing function ψ in
  such a way that ψ (û)∈ R(g_1)=R. Let us
  assume for the moment that this task has been accomplished. Then,
  from Lemma <ref>, it follows that ψ(û)
  is the unique maximizer of the functional L, whence we must have
  ĝ=ψ(û), which is the desired result.

  We now come to the issue of extending ψ_S. This is done in two
  steps. The first step is to show that û attains its largest
  values on S(ĝ). To this end, it suffices to prove the
  following inequality:

    α≡inf_S(ĝ)û≥sup_S(ĝ)^cû≡β,


where S(ĝ)^c denotes the complement of S(ĝ). We
prove (<ref>) by contradiction. So, let us suppose that
α<β. Hence, there exist constants γ,δ, and sets
A⊆ S(ĝ), B⊆ S(ĝ)^c, such that
β>γ>δ>α, and:


    {1ex1.2[     û≤δ  on  A,;     û≥γ on  B . ].


We may assume that |A|=|B|, otherwise we consider subsets of A and
B. Let η:A→ B be a measure preserving
bijection.[Such a map exists. See,
  , <cit.>.] Next, we define a new
function g as follows:


    g(x)
      =
      {1ex1.2[         ĝ(x) x∈ (A∪ B)^c,;     ĝ(η (x))        x∈ A,;   ĝ(η^-1(x))        x∈ B. ].


Clearly g is a rearrangement of ĝ. Since ĝ∈ R, it follows from Lemma <ref>
that g∈ R. Thus:


    ∫_Dgû^2  x-∫_Dĝû^2  x = ∫_A∪ Bgû^2  x-∫_A∪ Bĝû^2  x
      =∫_Bgû^2  x-∫_Aĝû^2  x 
    
       = ∫_Bĝ(η^-1(x))û^2  x-∫_Aĝû^2  x=∫_Aĝ(x)û^2(η(x))  x-∫_Aĝû^2  x
       ≥ (γ^2-δ^2)∫_Aĝ x>0,


which contradicts the maximality of ĝ.

In the second step, we give an explicit formula for the extended
function as follows:


    ψ̂ (t) =
      {1ex1.2[ ψ_S(t) t>α^2,;      0 t≤α^2, ].


where α is defined in (<ref>). Clearly, ψ̂
is increasing and ψ̂ (û^2)∈ R(g_1)=R.
Hence, by setting ψ(t) ψ̂(t^2) we derive
(<ref>). The proof of the theorem is completed.



  As mentioned earlier, in the special case of g_0=χ_E_0 with
  |E_0|=α, the minimization problem (<ref>)
  reduces to the one considered
  in <cit.>. So, ĝ=χ_Ê with |Ê|=α. Hence, from (<ref>)
  we deduce that Ê={û>γ}, for some
  γ>0. Whence, we derive the following boundary value problem:
  
    {1ex1.2[ -Δû+ûχ_{û>γ}=f(x)            in  D,;               û=0           on ∂ D. ].

  By setting U=û-γ, the differential equation in (<ref>) becomes:

    Δ U=(U+γ-f)χ_{U> 0}-fχ_{U ≤ 0}.


So, (<ref>) is an obstacle problem of type:


    Δ U=G(x)χ_{U > 0}-H(x)χ_{U ≤ 0},
 

where G≤ 0, because û≤ f, and H(x)≥ 0. Since G(x)+H(x)≥ 0,
we can apply the result of <cit.> to deduce that the free boundary has C1,1 regularity.






 §.§ Local minimizers



Even though ĝ in Theorem <ref> is a global minimizer,
is it possible for Φ to have non-global local minimizers
over R? The answer to this question is negative. To
prove this, we need a less restrictive version of Theorem 3.3 (iii)
in <cit.>, stated as follows:


  
  Let N: L^r(D)→ be weakly sequentially
  continuous, and let R= R(h_0) denote the rearrangement
  class generated by some h_0∈ L^r(D). Assume that for every pair
  (h_1,h_2)∈ R× R the
  following relation holds:
  
    lim_t→ 0^+ N(th_2+(1-t)h_1)-
            N(h_1)/t = ∫_D(h_2-h_1)   G  x,


  for some G∈ L^r^'(D). Suppose U is a
  strong neighborhood (relative to R) of ĥ∈
    R, for which we have:
  
    ∀ h∈ U:  N(ĥ)≤N(h).


  Then, ĥ minimizes the linear functional L(h) ∫_D h   G  x, relative to h∈ R.


Now we state our result concerning local minimizers.

  
  Let the hypotheses of Theorem <ref> hold. If g_1 and
  g_2 are two local minimizers of Φ(g) relative to g∈R,
  then g_1=g_2.



  For simplicity we set u_1  u_g_1 and u_2 
  u_g_2. Lemma <ref>, in conjunction
  with Lemma <ref> (<ref>), implies that g_1
  and g_2 are maximizers of the linear functionals:
  
    L_1(g) ∫_Dgu_1^2  x,

  and
  
    L_2(g) ∫_Dgu_2^2  x,


  relative to g∈ R, respectively. In particular, we
  infer:
  
    ∫_Dg_2u_1^2  x≤∫_Dg_1u_1^2  x   and  ∫_Dg_1u_2^2  x≤∫_Dg_2u_2^2  x.

  Thus, we obtain:
  0.5ex1.1
  
    2∫_Dfu_1  x-∫_D(|∇ u_1|^2+g_1u_1^2)  x   ≤   2∫_Dfu_1  x-∫_D(|∇ u_1|^2+g_2u_1^2)  x 
       ≤   2∫_Dfu_2  x-∫_D(|∇ u_2|^2+g_2u_2^2)  x 
       ≤    2∫_Dfu_2  x-∫_D(|∇ u_2|^2+g_1u_2^2)  x 
       ≤    2∫_Dfu_1  x-∫_D(|∇ u_1|^2+g_1u_1^2)  x,

  where the first and third inequalities are consequences of
  (<ref>), whereas the second and the fourth inequalities
  follow from (<ref>). From (<ref>) we see that all
  inequalities must in fact be equalities. This, in turn, implies that
  u_1=u_2, due to the uniqueness. Whence, we deduce g_1=g_2 as
  desired.







 §.§ Radial domain


Here we present our result regarding radial symmetry of the
optimizers. Note how, compared with similar results in the literature,
in our approach, such result may be obtained with minimal
technicalities:


  
  Suppose that f is radial and satisfies assumption A1. Then the solution of (<ref>) is radial and non-increasing.



  Let g denote the solution of (<ref>) and let R
  be a rotational map about the origin. Since f is radial, we infer
  u_g∘ R=u_g∘ R. Thus, Φ (g∘ R)=Φ (g), and
  g∘ R is also a solution of (<ref>). By
  uniqueness, we deduce g∘ R=g, for every rotational map
  R. Whence, g is radial, as desired. To prove that g is
  non-increasing, we observe that, since u=u_g is radial, we can
  write the equation in (<ref>) as:

    -(r^N-1u')'=r^N-1(f-gu).

Since f≥ v_f by A1, and g≤ 1 by assumption, we have
f-gu≥ v_f-u. Furthermore, v_f-u>0 by the proof of Lemma
<ref>. Hence,

    -(r^N-1u')'>0,   -r^N-1u'>0,    u'<0.


By Theorem <ref>, g=ψ(u), for some non-decreasing
ψ. As a result, g is non-increasing, as desired.






 §.§ Some remarks on maximization


In addition to the minimization problem (<ref>), one
can also consider the maximization problem:

    sup_g∈ RΦ (g).

Since Φ is weakly continuous and convex, Φ reaches its
maximum value at the extremal points of the convex set
ℛ (, the elements of ℛ). Hence,
problem (<ref>) is solvable (see Theorem 7 of
<cit.> or Remark 3.1 of
<cit.>). Moreover, if
the assumption A1 holds, along the same lines as in the proof of
Theorem <ref>, it can be shown that, if g̃ is a
maximizer, then:

    g̃=ψ̃(ũ),

almost everywhere in D, for some decreasing function ψ̃.
Here ũ=u_g̃, the solution of (<ref>) with
g=g̃.

Note that, for maximizers we do not have uniqueness in general.
However, we are going to prove that, in case D is a ball and f is
radially symmetric and non-increasing, any maximizer is radially
symmetric and non-decreasing, hence unique. Indeed, let
v=u_g̃_*, where g̃_* is the increasing Schwarz
symmetrization of g̃ (see <cit.>). For
simplicity, we write u instead of u_g̃. By
Lemma <ref> (<ref>):


    -1/2Φ (g̃)=1/2∫_D|∇
      u|^2  x+1/2∫_Dg̃ u^2  x-∫_Dfu  x
      ≥1/2∫_D|∇ u^*|^2  x+1/2∫_Dg̃
      u^2  x-∫_Dfu  x.


Now, by applying the Hardy-Littlewood inequality to the last two
integrals in (<ref>), keeping in mind that f=f^*, we obtain:


    -1/2Φ (g̃)≥1/2∫_D|∇
      u^*|^2  x+1/2∫_Dg̃_* u^*^2  x-∫_Dfu^*  x.

Recalling that v minimizes the functional


    I(w)=1/2∫_D|∇ w|^2  x+1/2∫_Dg̃_* w^2  x-∫_Dfw  x,


relative to w∈ H^1_0(D), we infer from (<ref>) that:

    -1/2Φ (g̃)≥1/2∫_D|∇ v|^2  x+1/2∫_Dg̃_* v^2  x-∫_Dfv  x=-1/2Φ (g̃_*).

As g̃ is maximal for Φ, then Φ (g̃_*)≤Φ
(g̃), which together with (<ref>), (<ref>) and
(<ref>) yield:

    ∫_D|∇ u|^2  x=∫_D|∇ u^*|^2  x.



Thus, from
Lemma <ref> (<ref>), we see
that u^-1(α, ∞) is a ball for every 0≤α<M=
sup u. We now proceed to show that u=u^*. Recalling
Lemma <ref> (<ref>), it
suffices to verify that the set {x∈ D: ∇ u=0, 0<u(x)<M}
is measure zero. To this end, consider x_0∈ D, and set S {u≥ u(x_0)}. We know that S is a disk (ball), and by
continuity of u, x_0∈∂ S⊆{u=u(x_0)}.  So we can
apply the Hopf lemma (see,
, <cit.>), and deduce
that ∂ u/∂ν(x_0)<0, where ν denotes the
unit outward normal vector to ∂ S at x_0. Whence, in
particular, ∇ u(x_0)≠ 0. Thus, in fact, {x∈ D: ∇
u=0, 0<u(x)<M} is empty, so its measure is zero, as desired. This
implies u=u^*, and by (<ref>), g̃=ψ̃(u^*)
almost everywhere in D. Since ψ̃ is decreasing, g̃ is radial and non-decreasing, as claimed.



  A consequence of (<ref>) is that the larger values of û are attained where ĝ is large. Whence, in case the set
  {ĝ=0} has positive measure, it will contain a layer around
  the boundary ∂ D, since û is continuous, and
  vanishes on ∂ D. Physically, this means that in the
    construction of a robust membrane one should use the material with
    least density near the boundary. The dual conclusion can be drawn
  similarly regarding the maximization problem (<ref>).



  Note that Theorem <ref> can be improved. Indeed, if
  D is Steiner symmetric with respect to a hyperplane l (see, , <cit.>), then ĝ (the solution of
  (<ref>)) will also be Steiner symmetric with respect to l. Of
  course, in this case, one needs to use the inequality:
  
    ∫_D|∇ u|^2  x≥∫_D|∇ u^♯|^2  x,


  instead of (<ref>), in which u^♯ stands for the Steiner
  symmetrization of u. A similar result can be obtained for the
  maximization problem (<ref>). Of course, for the
  maximization problem we do not necessarily have uniqueness of
  optimal solutions.








§ SHAPE OPTIMIZATION



In this section, we focus on the shape optimization variant of our
main problem, , the case where the generator g_0 is
two-valued. Thus, we consider the following boundary value problem:


    {1ex1.2[ -Δ u+(αχ_E +βχ_E^c)   u=f,                      in D,;                       u=0,                    on ∂ D, ].


in which, D is a smooth bounded domain in ^N, N ∈{ 2,3
}, f ∈ L^2(D) is a given non-negative function,
1≥α>β≥ 0, E is a measurable subset of D, and E^c
is the complement of E in D.[To see why the assumption 1
  ≥α is imposed, see Lemma <ref>
  [above]lem:estimate.]  Denoting the unique solution of
(<ref>) by u_E, we are interested in the
following minimization problem:

    inf_|E|=γ∫_Dfu_E  x,


where 0<γ<|D|. By Theorem <ref>, we know that, if f
satisfies A1, then (<ref>) has a unique
solution D̃⊂ D, with |D̃|=γ. Also, we
have D̃= {x∈ D:u_D̃(x)>c}, for some
positive c, which, in turn, implies:

    u_D̃(x)=c,  on ∂D̃.


Our aim is to analyze monotonicity and stability of solutions with
respect to the parameters α and γ. Analyses of this kind
are crucial for laying the foundation for computable analysis of shape
optimization problems such as (<ref>). 


In what follows, we keep the presentation succinct, and as such, many
of the claims will be listed with the proofs omitted. The interested
reader may refer to Sect. 3.3 of <cit.> for
the details of the omitted proofs. Nonetheless, we present the proofs
of a few of the more interesting cases.



 

 §.§ Monotonicity and stability results with respect to
   γ


 We know that, for each 0 < γ < |D|, the minimization problem
 (<ref>) has a unique solution. Now, consider
 0<γ_1,γ_2<|D|, and their corresponding unique solutions:
 
    D̃_γ_1={x∈ D: u_γ_1(x)>c_γ_1}  and  D̃_γ_2={x∈ D:
         u_γ_2(x)>c_γ_2},

 for some positive c_γ_1 and c_γ_2, where
 u_γ_1 and u_γ_2 satisfy:
 
    {1ex1.2[ -Δ u_γ_1+(αχ_D̃_γ_1
           +βχ_D̃_γ_1^c)u_γ_1=f,                                            in D,;                                         u_γ_1=0,                                          on ∂ D, ].

and

    {1ex1.2[ -Δ u_γ_2+(αχ_D̃_γ_2
          +βχ_D̃_γ_2^c)u_γ_2=f,                                           in D,;                                        u_γ_2=0,                                         on ∂ D. ].


We also restate the minimization problem
(<ref>), with γ as an input parameter:


    Ψ(γ) inf_|E|=γ∫_Dfu_E  x.




  
  If 0<γ_1<γ_2<|D|, then
  
  
  *  c_γ_1≥ c_γ_2.

  
  *  D̃_γ_1⊆D̃_γ_2.
    
  
  *  u_γ_1> u_γ_2 in D.
  
  
  



Since f is non-negative and non-trivial, the following is an easy
consequence of Proposition <ref> (<ref>).


  
  Ψ(γ) is a decreasing function on (0,|D|).



  
  If γ_1 tends to γ_2 in (0,|D|), then u_γ_1
  converges to u_γ_2 in C(D̅). Moreover,
  c_γ_1 converges to c_γ_2, where
  c_γ_1=u_γ_1(∂D̃_γ_1) and
  c_γ_2=u_γ_2(∂D̃_γ_2).





  
  Ψ(γ) is continuous on (0,|D|).


  From Corollary <ref> we infer that Ψ(γ) is
  differentiable almost everywhere. However, the following theorem
  shows that it is actually continuously differentiable on (0,|D|).


  
  Ψ(γ) is continuously differentiable on (0,|D|).
  Moreover:
  
    Ψ^'(γ)=-(α-β)c_γ^2  ,


  in which c_γ=u_γ(∂D̃_γ).


 Fix 0<γ_2<|D|, and let γ_1 increase to
  γ_2. We claim that
  Ψ(γ_1)-Ψ(γ_2)/γ_1-γ_2 converges
  to -(α-β)c_γ_2^2. From (<ref>) and (<ref>), we deduce

  
    {1ex1.2[ -Δ (u_γ_1-u_γ_2)+(αχ_D̃_γ_1
            +βχ_D̃_γ_1^c)(u_γ_1-u_γ_2)=
            -(α-β)u_γ_2(χ_D̃_γ_1-χ_D̃_γ_2)                                                                                                   in D;                                                                                          u_γ_1-u_γ_2=0                                                                                                on ∂ D. ].


  Multiplying the differential equation in (<ref>) by
  u_γ_1+u_γ_2, integrating the result over D,
  followed by an application of divergence theorem, in conjunction
  with D̃_γ_1⊆D̃_γ_2
  (Proposition <ref> (<ref>)) yields:

0.5ex1.1

    ∫_D(|∇ u_γ_1|^2-|∇ u_γ_2|^2 )  x   +   ∫_D(αχ_D̃_γ_1 +
      βχ_D̃_γ_1^c)(u_γ_1^2-u_γ_2^2)  x 
       =   -(α-β)∫_D u_γ_2(u_γ_1+
      u_γ_2)(χ_D̃_γ_1-χ_D̃_γ_2)  x
       =    (α-β)∫_D u_γ_2(u_γ_1+u_γ_2)χ_D̃_γ_2∖D̃_γ_1  x
       =
         
      (α-β)∫_D̃_γ_2∖D̃_γ_1
      u_γ_2(u_γ_1+u_γ_2)  x.


Furthermore, from (<ref>), (<ref>), and (<ref>), we
deduce:


    Ψ(γ_1)-Ψ(γ_2)=∫_D u_γ_1f  x-∫_D u_γ_2f  x 
    
       = [∫_D|∇ u_γ_1|^2   x +∫_D(αχ_D̃_γ_1+ βχ_D̃_γ_1^c)u_γ_1^2  x] -[∫_D|∇
        u_γ_2|^2   x  +∫_D(αχ_D̃_γ_2+
        βχ_D̃_γ_2^c)u_γ_2^2  x] 
    
      = ∫_D(|∇ u_γ_1|^2-|∇ u_γ_2|^2 )  x
      +
      ∫_D(αχ_D̃_γ_1+βχ_D̃_γ_2^c)(u_γ_1^2-u_γ_2^2)  x + ∫_D̃_γ_2∖D̃_γ_1(β
      u_γ_1^2-α u_γ_2^2)  x 
    
     = (α-
      β)∫_D̃_γ_2∖D̃_γ_1
      u_γ_2(u_γ_1+u_γ_2)  x-β∫_D̃_γ_2∖D̃_γ_1(u_γ_1^2-u_γ_2^2)  x +∫_D̃_γ_2∖D̃_γ_1(β u_γ_1^2-α
      u_γ_2^2)  x 
    
      = (α-β)∫_D̃_γ_2∖D̃_γ_1 u_γ_2u_γ_1  x.


where we have used the fact that D̃_γ_1⊆D̃_γ_2 in the third and fourth equality, and also
applied (<ref>) in the fourth equality. By using (<ref>)
and the fact that
|D̃_γ_2∖D̃_γ_1|=γ_2-γ_1,
we calculate:

0.5ex1.3

    |Ψ(γ_1)-Ψ(γ_2)/γ_1-γ_2-[-(α-β)c_γ_2^2]|
         =   α-β/γ_2-γ_1|∫_D̃_γ_2∖D̃_γ_1 (u_γ_2u_γ_1-
        c_γ_2^2)  x|
       ≤   
      (α-β)u_γ_2u_γ_1-c_γ_2^2
      _∞,D̃_γ_2∖D̃_γ_1.


By (<ref>) and Proposition <ref> (<ref>),
in D̃_γ_2∖D̃_γ_1 we have
c_γ_2< u_γ_2<u_γ_1≤ c_γ_1. So, by
applying Theorem <ref> we infer:


    u_γ_2u_γ_1-c_γ_2^2
      _∞,D̃_γ_2∖D̃_γ_1≤ |c_γ_1^2-c_γ_2^2|,


which converges to zero. From (<ref>), we obtain the desired
result.

Similarly, when γ_1 decreases to γ_2, the ratio
Ψ(γ_1)- Ψ(γ_2)/γ_1-γ_2 converges
to -(α-β)c_γ_2^2. By Theorem <ref>
we know that c_γ is continuous with respect to γ. Hence,
we infer that Ψ(γ) is continuously differentiable with
Ψ^'(γ)=-(α-β)c_γ^2 on (0,|D|).






 §.§ Monotonicity and stability results with respect to α



Assume that 0≤β<α_1,α_2≤ 1. For each of α_1
and α_2, the minimization problem (<ref>) has a unique
solution, which we denote by D̃_α_1 and
D̃_α_2, respectively. We know that
|D̃_α_1|=|D̃_α_2|=γ, and:


    D̃_α_1={x∈ D: u_α_1(x)>c_α_1}  and  D̃_α_2=
      {x∈ D: u_α_2(x)>c_α_2},


for some positive c_α_1 and c_α_2, where
u_α_1 and u_α_2 satisfy:


    {0.5ex1.2[ -Δ
          u_α_1+(α_1χ_D̃_α_1 +
          βχ_D̃_α_1^c)u_α_1=f,                                                    in D,;                                                 u_α_1=0,                                                  on ∂ D, ].


and


    {0.5ex1.2[ -Δ
          u_α_2+(α_2χ_D̃_α_2 +
          βχ_D̃_α_2^c)u_α_2=f,                                                    in D,;                                                 u_α_2=0,                                                  on ∂ D. ].








































This time, we restate the minimization problem
(<ref>), with α as an input parameter:


    Ψ(α) inf_|E|=γ∫_Dfu_E,α  x=∫_Dfu_α  x.





  If 0<α_1<α_2≤ 1, then:

  
  
  *  c_α_1> c_α_2.
  
  * u_α_1>u_α_2 in D.
  
  * D̃_α_1∩D̃_α_2≠∅.
  




  
  If β>0 and α_1 converges to α_2 in (β,1],
  then |D̃_α_1▵D̃_α_2|
  converges to zero.



 Fix β<α_2≤1 and let α_1 increase to
  α_2.  We claim that
  |D̃_α_1▵D̃_α_2| converges
  to zero. First, let us introduce the following auxiliary boundary
  value problem

  
    {0.5ex1.3[ -Δû_α_1+(α_1χ_D̃_α_2 +
            βχ_D̃_α_2^c)û_α_1=f                                              in D,;                                            û_α_1=0                                            on ∂ D. ].


From (<ref>) and (<ref>), we deduce:


    {0.5ex1.3[ -Δ
          (û_α_1-u_α_2)+
          (α_1χ_D̃_α_2 +βχ_D̃_α_2^c)(û_α_1-u_α_2)=
          (α_2-α_1)u_α_2χ_D̃_α_2                                                                                               in D,;                                                                                       û_α_1-u_α_2=0                                                                                             on ∂ D. ].


Since α_2>α_1, we infer that
(α_2-α_1)u_α_2χ_D̃_α_2 is
non-negative.  So, by applying the strong maximum principle to
(<ref>), we obtain û_α_1>u_α_2 in
D. Furthermore, by (<ref>), we have:


    D̂_α_1 = {x∈ D: û_α_1(x)>c_α_2}⊇{x∈ D:
      u_α_2(x)>c_α_2}=D̃_α_2.


Multiplying the differential equation in (<ref>) by
û_α_1-u_α_2, integrating the result over D,
followed by an application of divergence theorem yields:

0.5ex1.3

    ∫_D|∇(û_α_1-u_α_2)|^2 dx   +   ∫_D(α_1χ_D̃_α_2 +
      βχ_D̃_α_2^c)(û_α_1-u_α_2)^2dx
       =   (α_2-α_1)∫_D u_α_2(û_α_1-
      u_α_2)χ_D̃_α_2dx
       ≤    (α_2-α_1)u_α_2_4û_α_1
        -u_α_2_4|D̃_α_2|^1/2
       ≤    C(α_2-α_1)u_α_2_H_0^1(D)û_α_1
        -u_α_2_H_0^1(D)|D̃_α_2|^1/2,


where we have used general Hölder's inequality in the first
inequality, and Sobolev embedding theorem in the second
inequality. Since the second term of the first line of (<ref>)
is non-negative, we obtain:


    û_α_1
        -u_α_2_H_0^1(D)≤
      C(α_2-α_1)u_α_2_H_0^1(D)
      |D̃_α_2|^1/2.


Noting that α_1 increases to α_2, we infer
û_α_1 converges to u_α_2 in H_0^1(D).  By
using elliptic regularity theory and Sobolev embedding theorem, we
infer û_α_1 converges to u_α_2 in
C(D̅). So, from (<ref>) and the fact that
|D̃_α_2|=γ, in conjunction with Lemma
<ref>, we deduce that
|D̂_α_1∖D̃_α_2| decreases to zero,
and

    |D̂_α_1| →γ^+.


On the other hand, from (<ref>) and (<ref>), we have:


    -Δ
      (u_α_1-û_α_1)+(α_1χ_D̃_α_1
      +
      βχ_D̃_α_1^c)(u_α_1-û_α_1)
      =(α_1-β)û_α_1
      (χ_D̃_α_2∖D̃_α_1-χ_D̃_α_1∖D̃_α_2)  in  D,


with u_α_1-û_α_1=0 on ∂ D. Now, let us
introduce the following subsets of D:

    {0.5ex1.3[                                      Ê                                        {x∈ D:u_α_1(x)-û_α_1(x)≤ c_α_1-c_α_2},;                                     F̂                                        {x∈ D:u_α_1(x)-û_α_1(x)> c_α_1-c_α_2}. ].


Using (<ref>) and (<ref>), we infer
D̃_α_1∖D̂_α_1⊆F̂ and
D̂_α_1∖D̃_α_1⊆Ê. Moreover, by (<ref>), we have F̂=
(Ê)^c⊆(D̂_α_1∖D̃_α_1)^c⊆
(D̃_α_2∖D̃_α_1)^c. So,
(<ref>) leads to:


    -Δ
      (u_α_1-û_α_1)+(α_1χ_D̃_α_1
      +
      βχ_D̃_α_1^c∩D̃_α_2^c)(u_α_1-û_α_1)
      =-(α_1-β)û_α_1χ_D̃_α_1∖D̃_α_2  in F̂⊆(D̃_α_2∖D̃_α_1)^c.


Since c_α_1> c_α_2 (by
Proposition <ref> (<ref>)), we have
u_α_1-û_α_1=c_α_1-c_α_2>0 on
∂F̂. By applying the maximum principle to
(<ref>), we deduce u_α_1-û_α_1≤
c_α_1-c_α_2 in F̂. Recalling the definition of
F̂, we have F̂=∅. Since
D̃_α_1∖D̂_α_1⊆F̂, we
infer D̃_α_1∖D̂_α_1=∅,
 D̃_α_1⊆D̂_α_1. So, from
(<ref>) and the fact that |D̃_α_1|=γ, we
deduce |D̂_α_1∖D̃_α_1| decreases
to zero. Furthermore, recalling that
|D̂_α_1∖D̃_α_2| decreases to zero,
from (<ref>) we have:


    |D̃_α_1▵D̃_α_2|=|(D̃_α_1∖D̃_α_2)∪
    (D̃_α_2∖D̃_α_1)|≤
    |D̂_α_1∖D̃_α_2|+
    |D̂_α_1∖D̃_α_1| → 0^+,


when α_1 increases to α_2 as desired. Similarly, when
α_1 decreases to α_2, with β<α_2<1, we will
have |D̃_α_1▵D̃_α_2|
converging to zero. This completes the proof.




 
If β>0 and α_1 converges to α_2 in (β,1],
then u_α_1 converges to u_α_2 in C(D̅).



 If β>0 and α_1 converges to α_2 in (β,1], then Ψ(α_1) converges to
Ψ(α_2).






§ NUMERICAL SIMULATION



Numerical algorithms for solving rearrangement optimization problems
have appeared in the literature (See,
, <cit.>). As
there are no non-global local minima for problem
(<ref>), a simple gradient descent algorithm
suffices. Thus, we do not discuss the details of the algorithm here.

Nonetheless, we highlight a few issues regarding numerical simulation
of the problem (<ref>). It is clear from the
variational formulation (<ref>) that the
optimization problem (<ref>) is a minmax one. Speeding
up algorithms for rearrangement problems of this kind requires dealing
with certain heuristics, which are discussed in detail by Kao and
Su <cit.>.

The optimization problem (<ref>) of the current paper
should be contrasted with (say) the optimal harvesting problem
of <cit.>, or the steady
vortex problem considered
in <cit.>. Here are two major
differences:



  * Whereas the steady vortex and optimal harvesting problems can
  have uncountably many local optima and saddle points—which may
  only be partially overcome through the use of randomized
  algorithms <cit.>—problem
  (<ref>) has no non-global local minima.

  * On the other hand, the maxmax nature of the steady vortex
  problem and the minmin nature of the optimal harvesting problem
  provide for highly efficient algorithms that generate optimizing
  sequences. For problem (<ref>), however, careful use
  of heuristics is needed.



Using an approach similar to that
of <cit.>, we have
implemented an algorithm for the shape optimization problem
(<ref>). Figure <ref> (generated
by MATLAB®) illustrates one of our
monotonicity results, as stated in Proposition <ref>
(<ref>).









§ PROOF OF LEMMA <REF>





  * We follow the ideas
  in <cit.> (also,
  see <cit.>). Let {g_n}⊆R and g ∈R, such
  that g_n⇀ g in L^2(D). For simplicity, let us
  set u_n  u_g_n and u  u_g. We have:
  
  
    {1ex1.2[ -Δ u_n+g_nu_n=f           in D,;           u_n=0         on ∂ D. ].

  
  Multiplying the differential equation in (<ref>) by u_n, and
  integrating the result over D, yields
  
    ∫_D|∇
        u_n|^2  x+∫_Dg_nu_n^2  x=∫_Dfu_n  x.

  
  From Lemma <ref>, we know that g_n are
  non-negative. Therefore (<ref>) implies
  
  
    ∫_D|∇ u_n|^2  x≤∫_Dfu_n  x.
 By applying Hölder's inequality and the
  Poincaré inequality to the right hand side of
  (<ref>) we obtain
  
  
    ∫_D|∇ u_n|^2  x≤ Cf_2
          u_n_H_0^1(D),

  
  in which C is a positive constant. Whence, {u_n} is a
  bounded sequence in H_0^1(D). This in turn implies existence of a
  subsequence of {u_n}, still denoted {u_n}, and w∈
  H^1_0(D), such that:
  
  
    u_n⇀
        w   in H^1_0(D)     and     u_n→ w   in L^2(D).

  
  Let us prove that w=u, where u is the solution of
  
  
    {1ex1.2[ -Δ u+gu=f     in D,;       u=0   on ∂ D. ].

  Indeed, by (<ref>) we have
  
    ∫_D∇ u_n·∇ϕ  x+∫_Dg_nu_nϕ x=∫_Df
      ϕ x,    ∀ϕ∈ C^∞_0(D).
 Since
  u_n⇀ w in H^1_0(D), g_n⇀ g in
  L^2(D), and u_n→ w strongly in L^2(D), from the latter
  equation we find
  
    ∫_D∇ w·∇ϕ  x+∫_Dgw ϕ x=∫_Df
        ϕ x,    ∀ϕ∈ C^∞_0(D).

  
  This means that w is a solution of (<ref>), and by uniqueness,
  we must have w=u.  To prove (<ref>), we
  observe that
  
    |Φ(g_n)-Φ(g)|=|∫_Df(u_n-u)  x|≤
        ||f||_2    ||u_n-u||_2

  
  which together with the fact that lim_n →∞||u_n-u||_2 = 0
  implies (<ref>).


  * Let h,g∈R, 0<t<1, and
  ξ_t=th+(1-t)g. For v∈ H^1_0(D), we have
  
    2∫_Dfv  x-∫_D|∇
        v|^2  x - ∫_Dξ_tv^2  x  =  
    
        t(2∫_Dfv  x-∫_D|∇ v|^2  x -∫_Dhv^2  x) 
    
        + (1-t)(2∫_Dfv  x-∫_D|∇ v|^2  x - ∫_Dgv^2  x)

  
  By taking the supremum of (<ref>) with respect to v∈
  H^1_0(D), we obtain
  
    Φ(th+(1-t)g)≤ tΦ(h)+(1-t)Φ(g).

  This proves the convexity of Φ.  We now show, by
  contradiction, that Φ is in fact strictly convex. To this
  end, we assume that there exists t∈(0,1) such that
  Φ(th+(1-t)g)=t  Φ(h)+(1-t)Φ(g). For simplicity, we
  use u_t in place of u_th+(1-t)g. So, we have:
  
    2∫_Dfu_t  x-∫_D|∇
        u_t|^2  x - ∫_Dξ_tu_t^2  x   = 
    
        t(2∫_Dfu_h  x-∫_D|∇ u_h|^2  x -∫_Dhu_h^2  x)
    
        +  (1-t)(2∫_Dfu_g  x-∫_D|∇ u_g|^2  x -
          ∫_Dgu_g^2  x).

  
  From (<ref>), we deduce the following equations:
  
  
    2∫_Dfu_h  x-∫_D|∇ u_h|^2  x -∫_Dhu_h^2  x =
        2∫_Dfu_t  x-∫_D|∇ u_t|^2  x-∫_Dhu_t^2  x,

  and
  
    2∫_Dfu_g  x-∫_D|∇
        u_g|^2  x -∫_Dgu_g^2  x =
        2∫_Dfu_t  x-∫_D|∇ u_t|^2  x-∫_Dgu_t^2  x.

  From the maximality of u_h coupled with (<ref>), we infer
  u_h=u_t. Similarly, from the maximality of u_g and
  (<ref>), we find u_g=u_t.  Hence, u_t=u_h=u_g. On the
  other hand, from the differential equations 
    -Δ
      u_h+hu_h=f,  a.e. in  D,
 and
  
    -Δ u_g+gu_g=f,  a.e. in  D,

  we infer (h-g)u_h=0, almost everywhere in D. Since u_h is
  positive by the strong maximum principle, we must have h=g
  almost everywhere in D. Therefore, the strict convexity is
  proved.


  * For simplicity, we set u_t  u_ξ_t. We know that:
  
    {1ex1.2[ -Δ u_t+ξ_tu_t=f           in D,;           u_t=0         on ∂ D, ].

  and
  
    {1ex1.2[ -Δ u+gu=f     in D,;       u=0   on ∂ D. ].

  From (<ref>) and (<ref>), we obtain:
  
    -Δ (u_t-u)+g(u_t-u)=gu_t-ξ_tu_t=(g-ξ_t)u_t.

  
  Multiplying (<ref>) by u_t+u, and integrating the result
  over D, we get:
  
  
    ∫_D|∇ u_t|^2  x-∫_D|∇ u|^2  x +∫_D gu_t^2  x -
        ∫_Dgu^2  x 
    
        = ∫_D(g-ξ_t)u_t(u_t+u)  x
        = -t∫_D(h-g)u_t(u_t+u)  x.

  
  From (<ref>), we derive
  Φ(ξ_t)-Φ(g)=-t∫_D(h-g)u_tu  x, which in turn implies:
  
  0.5ex1.1
  
    Φ(ξ_t)-Φ(g)+t∫_D(h-g)u^2  x   =   -t∫_D(h-g)u_tu  x+t∫_D(h-g)u^2  x
        
       =   -t∫_D(h-g)(u_t-u)u  x.

  By applying Hölder's inequality to the right hand side of (<ref>), we find
  
    |Φ(ξ_t)-Φ(g)+t∫_D(h-g)u^2  x|≤ th-g_∞u_t-u_2u_2.

  Since ξ_t⇀ g weakly in L^2(D) (and even strongly),
  by the proof of part (<ref>), we have
  ||u_t-u||_2→ 0 as t→ 0. Hence, dividing by t in (<ref>)
  and letting t→ 0 we get the desired result. 







§ CONCLUDING REMARKS




In the main result of the current paper, ,
Theorem <ref>, we proved existence and uniqueness of
solutions for an optimization problem arising in construction of
robust membranes, with no restriction on the number of materials
used. This is yet another witness to the power and elegance of the
theory behind optimization of convex functionals over rearrangement
classes, as laid out by Burton <cit.>. Although the theory
was originally devised for studying vortex rings, , in the
context of fluid dynamics, ever since its introduction, there has been
a steady flow of contribution to the theory and its applications, in
fluid
mechanics <cit.>,
finance <cit.>,
free boundary problems <cit.>,
population biology <cit.>,
and eigenvalue problems <cit.>, to name
a few.

For the particular problem considered in the current paper, we showed
that there cannot be any non-global local optima
(Theorem <ref>). This has to be contrasted with other
rearrangement optimization problems where local optima and saddle
points
abound <cit.>. Furthermore,
we managed to deepen our understanding of the problem through some
stability results, which are, very difficult to prove, or even
formulate, in the presence of symmetry breaking, such as those
occurring
in <cit.>.













plain





10

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