Monotonicity and stability of optimal solutions of a minimization problem

Commun Nonlinear Sci Numer Simulat 25 (2015) 94–101

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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Monotonicity and stability of optimal solutions of a minimization problem Yichen Liu ⇑, Behrouz Emamizadeh Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Dushu Lake Higher Education Town, Suzhou, China

a r t i c l e

i n f o

Article history: Received 5 April 2013 Received in revised form 10 November 2014 Accepted 28 January 2015 Available online 7 February 2015 Keywords: Membrane Displacement Optimization Optimal solutions Tangent cones Optimality conditions Existence Uniqueness Monotonicity Stability

a b s t r a c t This paper is concerned with a minimization problem modeling the minimum displacement of an isotropic elastic membrane subject to a vertical force such as a load distribution. In addition to proving existence and uniqueness of optimal solutions, we show that these solutions are monotone and stable, in a certain sense. The main mathematical tool used in the analysis is the tangent cones from convex analysis, which helps to derive the optimality condition. Our results are compatible with physical expectations. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction In this paper, we study the following minimization problem:

inf Uðf Þ ¼

f 2Aa

Z D

fuf dx;

ð1:1Þ

  R where Aa ¼ f 2 L1 ðDÞ : 0 6 f 6 1; D f ðxÞdx ¼ a , for a given 0 < a < jDj. Here, uf denotes the unique solution of the following Poisson problem:



Du ¼ f ðxÞ in D; u¼0

on @D;

ð1:2Þ

where D is a smooth bounded domain in R2 . Physically, problem (1.2) models the vibration of an isotropic elastic membrane, fixed around the boundary, and subjected to a vertical force f ðxÞ (such as a load distribution). The function u stands for the displacement of the membrane from the rest position. The quantity Uðf Þ as defined in (1.1) measures the total displacement of the membrane. The dependence of U on ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (B. Emamizadeh). http://dx.doi.org/10.1016/j.cnsns.2015.01.016 1007-5704/Ó 2015 Elsevier B.V. All rights reserved.

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f is justified since by changing the force f the value of U naturally changes. We confine ourselves to forces described by Aa . In other words, the admissible forces are those that take values in ½0; 1, and have fixed total strength, designated by a. Note that the condition a 2 ð0; jDjÞ is to ensure the set Aa is not trivial. The minimization problem (1.1) implies that we are interested in the minimum value of the total displacement given that the forces applied to the membrane are selected from Aa . We are particularly interested in force functions that achieve the minimum. Such forces are called optimal solutions of (1.1). Indeed, we shall prove that optimal solutions exist. In fact, we will show that there exists a unique optimal solution. Our next result is that the optimal solution has a distinguished characterization; namely, it is of bang-bang type. Such name is referred to two-valued functions. In our case, the optimal solution will turn out to be a f0; 1g-valued, hence a characteristic function. Another feature of the optimal solution is that its support contains a layer around the boundary of D, which is expected from the physical point of view. Indeed, a force acting at a location near the boundary, noting that the membrane is held fixed at the boundary, will result in small displacement in contrast to when the same amount of force is applied to points which are located far from the boundary. To prove existence of optimal solutions for (1.1), we use a tool from convex analysis; namely, the tangent cones. It turns out that due to Aa being well structured, the tangent cone to Aa has a very convenient characterization. This characterization in conjunction with the convexity of Aa and / pave the way toward derivation of necessary and sufficient conditions for a function to be an optimal solution of (1.1). The method of tangent cones was recently used in [9], where the authors investigated a shape optimization problem. Surely, this method can also be applied to many other optimization problems, however, we should warn the readers that the method has its limitations. For example, in [3,4], the authors explore the possibility of designing a membrane, fixed at the boundary, and made out of two materials, so that the corresponding frequency is maximal. This is shape optimization problem to which the method of tangent cones can certainly be applied. However, if we look at the same problem allowing three or more materials used in the design then the method of tangent cones can no longer be accessible. After addressing the existence and uniqueness of optimal solutions, we present two monotonicity results, the first of which is physically quite interesting and in compliance with expectation. More precisely, we shall prove that by increasing the value of a, the support of the corresponding optimal solution increases in the sense of nested sets. The second monotonicity result is intriguing due to its physical interpretation that, the maximal distance from the rest position of the optimal solution is increasing with respect to a. In that section, we shall utilize some techniques from [6]. The last section of this paper will be allocated to showing that the optimal solutions of (1.1) are stable, see Theorem 5.1. Let us mention that Burton and McLeod [2] amongst other things studied a similar optimization problem to (1.1). They used the same functional as Uðf Þ, but considered a different admissible set. The admissible set, used in [2], was a rearrangement class, a set comprising functions which are equi-measurable with a given function. In that paper, the authors address existence and uniqueness of optimal solutions in general domains, and in radial domains in particular. 2. Preliminaries This section gathers the background for the sections to follow. Henceforth, we denote by j  j the Lebesgue measure in R2 . We begin with the definition of tangent cones. Definition 2.1. Let X be a normed linear space and C a nonempty set of X. The inner (intermediate, or derivable) tangent cone of C at a, denoted T 0C ðaÞ, is defined as follows: v 2 T 0C ðaÞ if and only if for each tn # 0 there exists a sequence fv n g1 n¼1 in X satisfying (i) limn!1 v n ¼ v , (ii) a þ t n v n 2 C; 8n 2 N. The following two lemmata are useful for deriving the minimality conditions associated with problem (1.1). Lemma 2.1. Let C and X be as in Definition 2.1, U : X ! R a functional which is Gâteaux differentiable and Lipschitz continuous in an open set E containing C. If f is a minimizer of U in C, then

hU0 ðf Þ; hi P 0;

8h 2 T 0C ðf Þ;

ð2:1Þ 

0

where h ;  i denotes the pairing between X and X , the dual of X. Here, U ðf Þ stands for the Gâteaux derivative of U at f. Proof. Throughout the proof, k denotes the Lipschitz constant of U. We assume the assertion of the lemma is false, to derive ^ 2 T 0 ðf Þ with hU0 ðf Þ; hi ^ < 0. Fix a sequence tn # 0. From the definition, there exists a sequence a contradiction. So we can find h C ^ ^ and keeping in mind that d is negative, there exists fhn g in X such that hn ! h and f þ tn hn 2 C; 8n 2 N. Set d ¼ hU0 ðf Þ; hi, N 1 2 N such that

^  Uðf Þ Uðf þ t n hÞ tn

<

d ; 2

8n P N 1 :

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  d ^ we can find N 2 2 N such that N 2 P N 1 ; f þ tN h ^ 2 E and  ^ Since hn ! h, . In particular, we have hN2  h  <  2k 2

d 2

^  Uðf Þ < tN : Uðf þ tN2 hÞ 2

ð2:2Þ

On the other hand, U is Lipschitz continuous in E, hence

 

 

d 2

^ 6 ktN hN  h ^ <  tN : Uðf þ tN2 hN2 Þ  Uðf þ tN2 hÞ 2 2 2

ð2:3Þ

From (2.2) and (2.3), we deduce Uðf þ t N2 hN2 Þ < Uðf Þ. This, recalling f þ tN2 hN2 2 C, contradicts the minimality of f. h Lemma 2.2. Let C be a nonempty convex set in X, and U : X ! R a convex functional which is Gâteaux differentiable. If hU0 ðf Þ; hi P 0 for all h in T 0C ðf Þ, then f is a minimizer of U in C. Proof. To derive a contradiction, let us assume the assertion is false. So we can find ^f 2 C such that Uð^f Þ < Uðf Þ. Observe that ^f  f 2 T 0 ðf Þ because C convex. Indeed, pick an arbitrary sequence tn # 0, and set v n ¼ ^f  f ; 8n 2 N. Clearly, v n ! ^f  f and C

f þ tn ð^f  f Þ ¼ tn ^f þ ð1  t n Þf 2 C as desired. Now by the assumption, hU0 ðf Þ; ^f  f i P 0. However,

Uðf þ tð^f  f ÞÞ  Uðf Þ ð1  tÞUðf Þ þ t Uð^f Þ  Uðf Þ hU0 ðf Þ; ^f  f i ¼ lim 6 lim ¼ Uð^f Þ  Uðf Þ < 0 t#0 t#0 t t which is a contradiction. Therefore, f is a minimizer of U in C. h In order to determine the characteristics of tangent cones in Aa , it is convenient to introduce the following notation: Notation. For any function f 2 Aa , we define the sets D0 ; D and D1 as follows.

(i) D0 ¼ fx 2 D : f ðxÞ ¼ 0g, (ii) D ¼ fx 2 D : 0 < f ðxÞ < 1g, (iii) D1 ¼ fx 2 D : f ðxÞ ¼ 1g.

Lemma 2.3. If f 2 Aa , then the tangent cone of Aa at f consists of functions h 2 L1 ðDÞ such that (i)

R

hðxÞdx ¼ 0,     (ii) limn!1 vQ 0n h  ¼ 0,  1  þ (iii) limn!1 vQ 1n h  ¼ 0, D

1

þ



where h (resp. h ) is the positive (resp. negative) part of h; Q 0n ¼ fx 2 D : f ðxÞ 6 1=ng and Q 1n ¼ fx 2 D : f ðxÞ P 1  1=ng. Proof. See Proposition 2.1 in [1] and Proposition 4.5 in [5]. h Lemma 2.4. Let f 2 Aa . If h 2 T 0Aa ðf Þ, then

hðxÞ  0 a:e: in D0 ;

hðxÞ 6 0 a:e: in D1 :

Proof. Observe that D0 # Q 0n and D1 # Q 1n . Whence, the assertion readily follows from Lemma 2.3. h Definition 2.2. We say the graph of f has no significant flat sections provided

   x 2 R2 : f ðxÞ ¼ c  ¼ 0;

8c 2 Rþ :

3. Existence and uniqueness of optimal solutions This section is devoted to the minimization problem (1.1). But first, we need the following basic result regarding the energy functional U.

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Lemma 3.1. The functional U enjoys the following properties: (i) (ii) (iii) (iv)

U U U U

is is is is

weak⁄-continuous in L1 ðDÞ. strictly convex in Aa . Gâteaux differentiable; moreover, U0 ðf Þ can be identified with 2uf . Lipschitz continuous in Aa .

Proof. For (i), (ii) and (iii), see Lemma 2.1 in [11]. We proceed to prove part (iv). To this end, we multiply the differential equation in (1.2) by uf , integrate the result over D, and finally apply the divergence theorem to deduce

Z

jruf j2 dx ¼

Z

D

D

fuf dx:

ð3:1Þ

An application of the Hölder’s inequality to the right hand side of (3.1), coupled with the Poincaré inequality, leads to

Z D

  jruf j2 dx 6 C kf k2 uf H1 ðDÞ ; 0

where C is a positive constant. From the last inequality, we infer

  uf  1 6 C kf k : 2 H ðDÞ

ð3:2Þ

0

For all f and g in Aa , we have

 Z  Z  Z Z       jUðf Þ  UðgÞj ¼  ðf  gÞuf dx þ gðuf  ug Þdx 6  ðf  gÞuf dx þ  gðuf  ug Þdx D D D D     6 C kf  g k2 uf H1 ðDÞ þ C kg k2 uf  ug H1 ðDÞ 6 Cðkf k2 þ kg k2 Þkf  g k2 6 C kf  g k1 ; 0

0

where we have used (3.2) in the third inequality, and the definition of Aa in the last inequality. h The main result of this section is the following. Theorem 3.2. The minimization problem (1.1) has a unique solution ^f . Moreover, ^f is characterized as follows: ^f minimizes Uðf Þ relative to Aa if and only if: (i) jD j ¼ 0, (ii) u^f ðx0 Þ P u^f ðx1 Þ; 8ðx0 ; x1 Þ 2 D0  D1 . Indeed, ^f is a characteristic function which is equal to vn

u^f
o , where c ¼ max  u^ > 0. D f

Proof. The proof is based on the notion of tangent cones, a tool that was also considered in [9]. We begin by observing that, Aa is closed in L1 ðDÞ, and convex. Hence, Aa is weak⁄-closed. By Theorem 2.10.2 in [10], we infer Aa is in fact weak⁄-compact. On the other hand, by Lemma (3.1) (i), U is weak⁄-continuous, hence (1.1) is solvable. We postpone addressing the uniqueness to the end of the proof. n o Let ^f be a solution of (1.1) in Aa , and set Dn ¼ x 2 D : 1=n 6 ^f 6 1  1=n . We first prove u^f is constant on D . To this end, S   observe that D ¼ 1 n¼1 Dn , hence it suffices to prove u^f is constant on Dn . To derive a contradiction, suppose u^f is not constant on Dn , for some n. Thus, there exist two measurable sets x1 and x2 in Dn such that

jx1 j ¼ jx2 j and

Z

x1

u^f dx <

Z

x2

u^f dx:

ð3:3Þ

Now taking

8 x 2 x1 ; > <1 hðxÞ ¼ 1 x 2 x2 ; > : 0 x 2 ð x1 [ x2 Þ c ; which belongs to T 0Aa ð^f Þ (see Lemma 2.3), yields

hU0 ð^f Þ; hi ¼ 2

Z

D

u^f hdx ¼ 2

Z

x1

u^f dx  2

Z

x2

u^f dx < 0;

ð3:4Þ

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by (3.3). Clearly, (3.4) contradicts the optimality condition (2.1). Thus, u^f is constant on D . Next, from the differential equation (1.2), coupled with Lemma 7.7 in [8], we infer that the graph of u^f has no significant flat sections in D . Therefore, jD j ¼ 0, as desired. For part (ii), let us assume there exist two measurable sets x0 # D0 and x1 # D1 such that

jx0 j ¼ jx1 j and

Z

x0

u^f dx <

Z

x1

u^f dx:

ð3:5Þ

Next, we set:

8 x 2 x0 ; > <1 hðxÞ ¼ 1 x 2 x1 ; > : 0 x 2 ð x0 [ x1 Þ c ; which belongs to T 0Aa ð^f Þ. Similarly to the proof of part (i), the inequality in (3.5) leads to a contradiction of the optimality ^ ¼ a. By the elliptic regularity condition (2.1). Since jD j ¼ 0, we deduce that ^f must be a characteristic function v ^ , where jDj D

 Using theory, see for example [7], we infer u^f 2 H2 ðDÞ, thus, by the Sobolev embedding theorem, it follows that u^f 2 CðDÞ. n o ^ ¼ x 2 D : u^ ðxÞ 6 c , where part (ii), we deduce D f

c ¼ sup u^f ðxÞ ¼ inf u^f ðxÞ > 0: x2D0

x2D1

n o ^ ¼ x 2 D : u^ ðxÞ < c , where c ¼ max  u^ > 0. Clearly, we have u^f ¼ c on @D0 , which implies u^f ¼ c in D0 . Whence, D D f f Conversely, let us assume the pair ð^f ; u^f Þ satisfies (i) and (ii). Due to the continuity of u^f , we deduce

c ¼ sup u^f ðxÞ ¼ inf u^f ðxÞ > 0: x2D0

x2D1

Let us fix h in T 0Ac ð^f Þ. From Lemmata 2.3 and 2.4, we obtain

hU0 ð^f Þ; hi ¼ 2

Z

D

u^f hdx ¼ 2

Z

D0

u^f hdx þ 2

Z

D1

u^f hdx P 2

Z

c hdx þ 2 D0

Z D1

c hdx ¼ 2c

Z

hdx ¼ 0:

D

Therefore, we infer from Lemma 2.2 that ^f is a minimizer. Finally, we settle the issue of uniqueness. To this end, we assume ^f is a solution of (1.1). To derive a contradiction, let us assume ~f is another solution of (1.1). We set g ¼ 12 ð^f þ ~f Þ which belongs to Aa , because Aa is convex. Recalling that U is strict convex, it now follows that UðgÞ < Uð^f Þ, which contradicts the minimality of ^f . h Proposition 3.3. If D is simply connected and layer around @D.

vD^ is the unique minimizer of problem (1.1), then D^ is connected and contains a

n o ^ ¼ x 2 D : u^ ðxÞ < c , where c ¼ max  u^ > 0. This implies D ^ contains a layer around Proof. From Theorem 3.2, we infer D D f f  and u^ vanishes on @D. To prove D ^ is connected, we assume otherwise and derive a contradiction. So let us @D, since u^f 2 CðDÞ f ^ denoted D0 , such that the intersection of @D0 and @D is empty. Observe that, u^ ¼ c on @D0 . assume there is a component of D, f Recalling the Poisson problem (1.2), it follows that

(

Du^f ¼ 1 in D0 u^f ¼ c

ð3:6Þ

on @D0 :

Hence by the strong maximum principle, we deduce, from (3.6), u^f > c in D0 . This clearly contradicts the fact that u^f < c in D0 . h

4. Monotonicity results In this section, we address two monotonicity results related to problem (1.1). Let us fix some notation. Similarly to Aa , we define Ab as follows:

Ab ¼

  Z f 2 L1 ðDÞ : 0 6 f 6 1; f ðxÞdx ¼ b ; D

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where 0 < b < jDj. By Theorem 3.2, we know the two minimization problems

inf Uðf Þ and

f 2Aa

inf Uðf Þ;

f 2Ab

have unique solutions, which we denote them by

vD^ a and vD^ b , respectively. Furthermore, we have



^ a ¼ fx 2 D : ua ðxÞ < ca g and D ^ b ¼ x 2 D : ub ðxÞ < cb D



ð4:1Þ

for positive ca ¼ maxD ua and cb ¼ maxD ub , where ua and ub satisfy:



Dua ¼ vD^ a

in D;

ua ¼ 0

on @D

Dub ¼ vD^ b

in D;

ub ¼ 0

on @D:

ð4:2Þ

and

(

ð4:3Þ

Our first monotonicity result is the following. ^b # D ^ a. Theorem 4.1. If 0 < b 6 a < jDj, then D Proof. Let us introduce the following subsets of D:

    E ¼ ua  ub > ca  cb and F ¼ ua  ub 6 ca  cb : ^a n D ^ b , and ua  ub > ca  cb in D ^b n D ^ a . By the definition of E and F, it follows that From (4.1), we infer ua  ub < ca  cb in D

^a n D ^ b # F; D

ð4:4Þ

^b n D ^ a # E: D

ð4:5Þ

^ aÞ [ D ^ b . From the differential equations in (4.2) From (4.4), in conjunction with the fact that F ¼ D n E, we deduce E # ðD n D and (4.3), we obtain

^ aÞ [ D ^ b: Dðua  ub Þ ¼ vD^ a  vD^ b 6 0 in E # ðD n D On the other hand, we have ua  ub ¼ ca  cb on @E. Whence, by the weak maximum principle, ua  ub 6 ca  cb in E. Recall^b n D ^ a is empty as well. ing the definition of E, we can conclude E must be empty. Furthermore, it follows from (4.5) that D ^b # D ^ a , as desired. h Hence, D Our second monotonicity result is as follows. Theorem 4.2. If 0 < b 6 a < jDj, then cb 6 ca . Proof. From (4.2) and (4.3), it follows that

(

Dðua  ub Þ ¼ vD^ a  vD^ b

in D;

ua  ub ¼ 0

on @D:

ð4:6Þ

Multiplying the differential equation in (4.6) by ua  ub , integrating the result over D, followed by an application of divergence theorem, yields

Z D

  rðua  ub Þ2 dx ¼

Z D

ðvD^ a  vD^ b Þðua  ub Þdx ¼

Z ^ a nD ^b D

ðua  ub Þdx þ

Z ^ b nD ^a D

ðub  ua Þdx:

ð4:7Þ

^a n D ^ b , and ua  ub > ca  cb in D ^b n D ^ a. Similarly to the proof of Theorem 4.1, recalling (4.1), we infer ua  ub < ca  cb in D Thus, Eq. (4.7) leads to

Z

   ^a n D ^ b jðca  cb Þ þ jD ^b n D ^ a jðcb  ca Þ ¼ ðca  cb Þ jD ^a n D ^ b j  jD ^b n D ^ aj : rðua  ub Þ2 dx 6 jD

ð4:8Þ

D

^a n D ^ b j ¼ jD ^ a j  jD ^a \ D ^ b j and jD ^b n D ^ a j ¼ jD ^ b j  jD ^a \ D ^ b j, we infer Because jD

^a n D ^ b j  jD ^b n D ^ a j ¼ jD ^ a j  jD ^ b j: jD

ð4:9Þ

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Substituting (4.9) into (4.8), it follows that

Z

   ^ a j  jD ^ b j ¼ ðca  cb Þða  bÞ: rðua  ub Þ2 dx 6 ðca  cb Þ jD

D

Since the left hand side of the last equation is nonnegative, and the fact that b 6 a, we infer cb 6 ca . h 5. A stability result Before stating the result of this section, we fix some notation. Let

 Aan ¼

f 2 L1 ðDÞ : 0 6 f 6 1;

Z

 f ðxÞdx ¼ an ;

D

where 0 < an < jDj. Let

vD^ n denote the unique solution of the following minimization problem:

inf Uðf Þ:

f 2Aan

The symmetric difference of two sets E and F is denoted by E 4 F. Our stability result is the following: Theorem 5.1. Let ^ n 4 Dj ^ ! 0. jD

^ ¼ a. If an ! a, then v ^ ! v ^ in L1 ðDÞ. Moreover, vD^ denote the minimizer of problem (1.1), satisfying jDj Dn D

Proof. Since an ! a, we infer

Z  jan  aj ¼ 

D

vD^ n dx 

Z D

 

Z 

 

vD^ dx ¼  ðvD^ n  vD^ Þdx ¼ D

Z D

jvD^ n  vD^ jdx ! 0;

where the third equality in (5.1) holds as a result of Theorem 4.1. Hence, Since

Z D

ð5:1Þ

vD^ n ! vD^ in L1 ðDÞ.

^ n 4 Dj; ^ jvD^ n  vD^ jdx ¼ jD

^ n 4 Dj ^ ! 0. This completes the proof of the theorem. h we also deduce jD 6. Conclusion We considered a minimization problem modeling minimum displacement of an isotropic elastic membrane subject to a vertical force. We wrote the minimality condition, that optimal solutions satisfy, in terms of the tangent cone. The structure of the admissible set made it possible to characterize the tangent cone in a very accessible way. Using the minimality condition we proved existence of optimal solutions, and later showed uniqueness. We then proved two monotonicity results with respect to the parameter involved in the admissible set, and this was followed by a stability result. We anticipate the maximization version of our problem can also be handled using the method introduced in this paper. However, problems of minmax and maxmin types are unlikely to be applicable. One interesting issue is to determine the qualitative properties of ^ :¼ @fu^ < cg. The authors plan to address this in a follow up paper, where the blow up method in conthe free boundary @ D f

^ is C 1;a . The junction with the monotonicity formula is applied to prove that the optimal regularity of the free boundary @ D same question can be raised for the maximization problem as well, but the authors up to this point have not been successful to obtain any results in this direction. References [1] Bednarczuk E, Pierre M, Rouy E, Sokolowski J. Tangent sets in some functional spaces. Nonlinear Anal Ser A Theory Methods 2000;42(5):871–86. [2] Burton GR, McLeod JB. Maximisation and minimisation on classes of rearrangements. Proc R Soc Edinburgh Sec A 1991;119(3–4):287–300. [3] Chanillo S, Grieser D, Imai M, Kurata K, Ohnishi I. Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun Math Phys 2000;214(2):315–37. [4] Chanillo S, Grieser D, Kurata K. The free boundary problem in the optimization of composite membranes. Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), 61–81. Contemporary Mathematics, vol. 268. Providence, RI: American Mathematical Society; 2000. [5] Cominetti R, Penot J-P. Tangent sets of order one and two to the positive cones of some functional spaces. Appl Math Optim 1997;36(3):291–312. [6] Cuccu F, Jha K, Porru Giovanni G. Geometric properties of solutions to maximization problems. Electron J Differ Eqs 2003;71:1–8. [7] Evans LC. Partial differential equations. Graduate studies in mathematics, second ed., vol. 19. Providence, RI: American Mathematical Society; 2010.

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[8] Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Classics in mathematics. Berlin: Springer-Verlag; 2001 (reprint of the 1998 ed.). [9] Henrot A, Maillot H. Optimization of the shape and the location of the actuators in an internal control problem. Boll Unione Mat Ital Sez B Artic Ric Mat 2001;4(8):737–57 (no. 3). [10] Hille E, Phillips RS. Functional analysis and semi-groups. American mathematical society colloquium publications, rev. ed., vol. 31. Providence, RI: American Mathematical Society; 1957. [11] Marras M. Optimization in problems involving the p-Laplacian. Electron J Differ Eqs 2010(02):1–10.