Some optimization problems for nonlinear elastic membranes
aa r X i v : . [ m a t h . A P ] J un SOME OPTIMIZATION PROBLEMS FOR NONLINEAR ELASTICMEMBRANES
LEANDRO M. DEL PEZZO AND JULI ´AN FERN ´ANDEZ BONDER
Abstract.
In this paper we study some optimization problems for nonlinearelastic membranes. More precisely, we consider the problem of optimizing thecost functional J ( u ) = R ∂ Ω f ( x ) u d H N − over some admissible class of loads f where u is the (unique) solution to the problem − ∆ p u + | u | p − u = 0 in Ωwith |∇ u | p − u ν = f on ∂ Ω. Introduction
In this paper we analyze the following optimization problem: Consider a smoothbounded domain Ω ⊂ R N and some class of admissible loads A . Then we want tomaximize the cost functional J ( f ) := Z ∂ Ω f ( x ) u d H N − , for f ∈ A , where H d denotes the d − dimensional Hausdorff measure and u is the(unique) solution to the nonlinear membrane problem with load f (1.1) ( − ∆ p u + | u | p − u = 0 in Ω , |∇ u | p − ∂u∂ν = f on ∂ Ω . Here, ∆ p u = div( |∇ u | p − ∇ u ) is the usual p − Laplacian and ∂∂ν is the outer unitnormal derivative.These types of optimization problems have been considered in the literature dueto many applications in science and engineering, specially in the linear case p = 2.See for instance [5].In recent years, models involving the p − Laplacian operator with nonlinearboundary conditions have been used in the theory of quasiregular and quasiconfor-mal mappings in Riemannian manifolds with boundary (see [9, 19]), non-Newtonianfluids, reaction diffusion problems, flow through porus media, nonlinear elasticity,glaciology, etc. (see [1, 2, 3, 8]).We want to stress that our results are new, even in the linear case. But sinceour arguments are mainly variational, and for the sake of completeness, we decidedto present the paper in this generality.In this work, we have chosen three different classes of admissible functions A towork with. • The class of rearrangements of a given function f . • The (unit) ball in some L q . • The class of characteristic functions of sets of given surface measure.
This latter case is what we believe is the most interesting one and where ourmain results are obtained.For each of these classes, we prove existence of a maximizing load (in the respec-tive class) and analyze properties of these maximizers.The approach to the class of rearrangements follows the lines of [6], where asimilar problem was analyzed, namely, the maximization of the functional¯ J ( g ) := Z Ω gu d H N , where u is the solution to − ∆ p u = g in Ω with Dirichlet boundary conditions.When we work in the unit ball of L q the problem becomes trivial and we explicitlyfind the (unique) maximizer for J , namely, the first eigenfunction of a Steklov-likenonlinear eigenvalue problem (see Section 4).Finally we arrive at the main part of the paper, namely, the class of characteristicfunctions of sets of given boundary measure. In order to work within this class,we first relax the problem and work with the weak* closure of the characteristicfunctions (i.e. bounded functions of given L norm), prove existence of a maximizerwithin this relaxed class and then prove that this optimizer is in fact a characteristicfunction. Then, in order to analyze properties of this maximizer, we compute thefirst variation (or shape derivative) with respect to perturbations on the set wherethe characteristic function is supported.This approach for optimization problems has been used several times in theliterature. Just to cite a few, see [7, 12, 15] and references therein. Also, ourapproach to the computation of the first variation borrows ideas from [13].The paper is organized as follows. In Section 2 we include some preliminaryresults, some of which are well known but we choose to include them in orderto make the paper self contained. In Section 3 we study the problem when theadmissible class of loads A is the class of rearrangements of a given function f .In Section 4, we study the simpler case when A is the unit ball in L q . Finally, inSection 5, we analyze the case where A is the class of characteristic functions ofsets with given surface measure.2. Preliminaries
In this section we collect some well known results that will be used throughoutthe paper.2.1.
Results on rearrangements.
First, we recall some well known facts on re-arrangements that will be needed in Section 3.
Definition 2.1.
Suppose f : ( X, Σ , µ ) → R + and g : ( X ′ , Σ ′ , µ ′ ) → R + are mea-surable functions. We say f and g are rearrangements of each other if and onlyif µ ( { x ∈ X : f ( x ) ≥ α } ) = µ ′ ( { x ∈ X ′ : g ( x ) ≥ α } ) , ∀ α ≥ . Now, given f ∈ L p ( A ), where A ⊂ R N with H d ( A ) < ∞ , the set of allrearrangements of f is denoted by R f . Thus, for any f ∈ R f , we have H d ( { x ∈ A : f ( x ) ≥ α } ) = H d ( { x ∈ A : f ( x ) ≥ α } ) , ∀ α ≥ . We will need the following Lemma, the proof of which can be found in [4].
ONLINEAR ELASTIC MEMBRANES 3
Lemma 2.2.
Let f ∈ L p ( ∂ Ω) and v ∈ L p ′ ( ∂ Ω) such that f , v ≥ . Then thereexists ˆ f ∈ R f such that Z ∂ Ω ˆ f v d H N − = sup h ∈R f Z ∂ Ω hv d H N − . The following result can be easily deduced from [17] (Theorem 1.14 p.28).
Theorem 2.3 (Bathtub Principle) . Let (Ω , Σ , µ ) be a measurable space and let f be a real-valued, measurable function on Ω such that µ ( { x : f ( x ) > t } ) is finite forall t ∈ R . Let the number G > be given and define the class C of measurablefunctions on Ω by C = (cid:8) g : 0 ≤ g ( x ) ≤ for all x and Z Ω g ( x ) d µ = G (cid:9) . Then the maximization problem I = sup g ∈C Z Ω f ( x ) g ( x ) d µ is solved by (2.1) g ( x ) = χ { f>s } ( x ) + cχ { f = s } ( x ) , where s = inf { t : µ ( { f ≥ t } ) ≤ G } and cµ ( { f = s } ) = G − µ ( { f > s } ) . The maximizer given in (2.1) is unique if G = µ ( { f > s } ) or if G = µ ( { f ≥ s } ) . Results on differential geometry.
Now we state without proof some resultson differential geometry that will be used in the last section. The proof of theseresults can be found, for instance, in [14].
Definition 2.4 (Definition of the tangential Jacobian) . Let Ω ⊂ R N be a smoothopen set of R N . Let Φ be a C field over R N . We call the tangential Jacobian of Φ J τ (Φ) := | T [Φ ′ ] − ν | J (Φ) , where ν is the outer unit normal vector to ∂ Ω , Φ ′ denotes the differential matrix of Φ , J (Φ) is the usual Jacobian of Φ and T A is the transpose of the matrix A . The definition of the tangential Jacobian is suited to state the following changeof variables formula
Proposition 2.5.
Let f ∈ L (Φ( ∂ Ω)) . Then f ◦ Φ ∈ L ( ∂ Ω) and Z Φ( ∂ Ω) f d H N − = Z ∂ Ω ( f ◦ Φ) J τ (Φ) d H N − . Definition 2.6 (Definition of the tangential divergence) . Let W be a C vectorfield defined on R N . The tangential divergence of W over ∂ Ω is defined as div τ W := divW − h W ′ ν, ν i , where ν is the outer unit normal vector to ∂ Ω and h· , ·i is the usual scalar productin R N . L. DEL PEZZO AND J. FERN´ANDEZ BONDER
With these definitions, we have the following version of the divergence Theorem.
Theorem 2.7.
Let Ω be a bounded smooth open set of R N , D ⊂ ∂ Ω be a (relatively)open smooth set. Let W be a [ W , ( ∂ Ω)] N vector field. Then Z D div τ W d H N − = Z ∂ D h W , ν τ i d H N − + Z D H h W , ν i d H N − , where ν τ is the outer unit normal vector to D along ∂ Ω and H is the mean curvatureof ∂ Ω . Maximizing in the class of rearrangements
Given a domain Ω ⊂ R N (bounded, connected, with smooth boundary), first wewant to study the following problem(3.1) ( − ∆ p u + | u | p − u = 0 in Ω , |∇ u | p − ∂u∂ν = f on ∂ Ω . Here p ∈ (1 , ∞ ), ∆ p u = div( |∇ u | p − ∇ u ) is the usual p − Laplacian, ∂∂ν is the outernormal derivative and f ∈ L q ( ∂ Ω) with q > p ′ N ′ .We say u ∈ W ,p (Ω) is a weak solution of (3.1) if Z Ω |∇ u | p − ∇ u ∇ v + | u | p − uv d H N = Z ∂ Ω f v d H N − for all v ∈ W ,p (Ω).The restriction q > p ′ N ′ is related to the fact that p ′ N ′ = p ′∗ where p ∗ = p ( N − / ( N − p ) is the critical exponent in the Sobolev trace imbedding W ,p (Ω) ֒ → L r ( ∂ Ω). So, in order for that the right side of last equality to make sense for f ∈ L q ( ∂ Ω) we need v to belong to L q ′ (Ω). This is achieved by the restriction q ′ < p ∗ .It is a standard result that (3.1) has a unique weak solution u f , for which thefollowing equations hold(3.2) Z ∂ Ω f u f d H N − = sup u ∈ W ,p (Ω) I ( u ) , where I ( u ) = 1 p − n p Z ∂ Ω f u d H N − − Z Ω |∇ u | p + | u | p d H N o . Let f ∈ L q ( ∂ Ω), with q = p/ ( p − , and let R f be the class of rearrangementsof f . We are interested in finding(3.3) sup f ∈R f Z ∂ Ω f u f d H N − . Theorem 3.1.
There exists ˆ f ∈ R f such that J ( ˆ f ) = Z ∂ Ω ˆ f ˆ u d H N − = sup f ∈R f J ( f ) = sup f ∈R f Z ∂ Ω f u f d H N − , where ˆ u = u ˆ f . ONLINEAR ELASTIC MEMBRANES 5
Proof.
Let I = sup f ∈R f Z ∂ Ω f u f d H N − . We first show that I is finite. Let f ∈ R f . By H¨older’s inequality and the traceembedding we have Z Ω |∇ u f | p + | u f | p d H N ≤ C k f k L q ( ∂ Ω) k u f k W ,p (Ω) , then(3.4) k u f k W ,p (Ω) ≤ C ∀ f ∈ R f since k f k L q ( ∂ Ω) = k f k L q ( ∂ Ω) for all f ∈ R f . Therefore I is finite.Now, let { f i } i ≥ be a maximizing sequence and let u i = u f i . From (3.4) it isclear that { u i } i ≥ is bounded in W ,p (Ω), then there exists a function u ∈ W ,p (Ω)such that, for a subsequence that we still call { u i } , u i ⇀ u weakly in W ,p (Ω) ,u i → u strongly in L p (Ω) ,u i → u strongly in L p ( ∂ Ω) . On the other hand, since { f i } i ≥ is bounded in L q ( ∂ Ω), we may choose a subse-quence, still denoted by { f i } i ≥ , and f ∈ L q ( ∂ Ω) such that f i ⇀ f weakly in L q ( ∂ Ω) . Then I = lim i →∞ Z ∂ Ω f i u i d H N − = 1 p − i →∞ n p Z ∂ Ω f i u i d H N − − Z Ω |∇ u i | p + | u i | p d H N o ≤ p − n p Z ∂ Ω f u d H N − − Z Ω |∇ u | p + | u | p d H N o . Furthermore, by Lemma 2.2, there exists ˆ f ∈ R f such that Z ∂ Ω f u d H N − ≤ Z ∂ Ω ˆ f u d H N − . Thus I ≤ p − n p Z ∂ Ω ˆ f u d H N − − Z Ω |∇ u | p + | u | p d H N o . As a consequence of (3.2), we have that I ≤ p − n p Z ∂ Ω ˆ f u d H N − − Z Ω |∇ u | p + | u | p d H N o ≤ p − n p Z ∂ Ω ˆ f ˆ u d H N − − Z Ω |∇ ˆ u | p + | ˆ u | p d H N o = Z ∂ Ω ˆ f ˆ u d H N − ≤ I. Recall that ˆ u = u ˆ f . Therefore ˆ f is a solution to (3.3). This completes the proof. (cid:3) L. DEL PEZZO AND J. FERN´ANDEZ BONDER
Remark 3.2.
With a similar proof we can prove a slighter stronger result. Namely,we can consider the functional J ( f, g ) := Z Ω gu d H N + Z ∂ Ω f u d H N − , where u is the (unique, weak) solution to ( − ∆ p u + | u | p − u = g in Ω , |∇ u | p − ∂u∂ν = f on ∂ Ω , and consider the problem of maximizing J over the class R g × R f for some fixed g and f .We leave the details to the reader. Maximizing in the unit ball of L q In this section we consider the optimization problemmax J ( f )where the maximum is taken over the unit ball in L q ( ∂ Ω).In this case, the answer is simple and we find that the maximizer can be computedexplicitly in terms of the extremal of the Sobolev trace embedding.So, we let f ∈ L q ( ∂ Ω) , with q > p ′ N ′ , and k f k L q ( ∂ Ω) ≤
1, we consider the problem(4.1) sup f ∈ Lq ( ∂ Ω) k f k Lq ( ∂ Ω) ≤ Z ∂ Ω f u f d H N , where u f is the weak solution of(4.2) ( − ∆ p u + | u | p − u = 0 in Ω , |∇ u | p − ∂u∂ν = f on ∂ Ω . The restriction q > p ′ N ′ is the same as in the previous section.In this case it is easy to see that the solution becomes ˆ f = v q ′ − q ′ where v q ′ ∈ W ,p (Ω) is a nonnegative extremal for S q ′ normalized such that k v q ′ k L q ′ ( ∂ Ω) = 1and S q ′ is the Sobolev trace constant given by S q ′ = inf v ∈ W ,p (Ω) R Ω |∇ v | p + | v | p d H N (cid:0) R ∂ Ω | v | q ′ d H N − (cid:1) pq ′ . Furthermore ˆ u = u ˆ f = S /p − q ′ v q ′ . Observe that, as q ′ < p ∗ there exists an extremalfor S q ′ . See [11] and references therein.In fact J ( ˆ f ) = Z ∂ Ω ˆ f ˆ u d H N − = Z Ω |∇ ˆ u | p + | ˆ u | p d H N = 1 S p/ ( p − q ′ Z Ω |∇ v q ′ | p + | v q ′ | p d H N = 1 S / ( p − q ′ . ONLINEAR ELASTIC MEMBRANES 7
On the other hand, given f ∈ L q ( ∂ Ω), such that k f k L q ( ∂ Ω) ≤
1, we have J ( f ) = Z ∂ Ω f u f d H N − ≤ k f k L q ( ∂ Ω) k u f k L q ′ ( ∂ Ω) ≤ (cid:16) S q ′ Z Ω |∇ u f | p + | u f | p d H N (cid:17) /p = 1 S /pq ′ (cid:16) Z ∂ Ω f u f d H N − (cid:17) /p , from which it follows that J ( f ) ≤ S / ( p − q ′ . This completes the characterization of the optimal load in this case.5.
Maximizing in L ∞ Now we consider the problem(5.1) sup φ ∈ B Z ∂ Ω φu φ d H N − , where B := { φ : 0 ≤ φ ( x ) ≤ x ∈ ∂ Ω and R ∂ Ω φ d H N − = A } , for somefixed 0 < A < H N − ( ∂ Ω) , and u φ is the weak solution of(5.2) ( − ∆ p u + | u | p − u = 0 in Ω , |∇ u | p − ∂u∂ν = φ on ∂ Ω . This is the most interesting case considered in this paper.5.1.
Existence of optimal configurations.
In this case, we have the followingtheorem:
Theorem 5.1.
There exists D ⊂ ∂ Ω with H N − ( D ) = A such that Z ∂ Ω χ D u D d H N − = sup φ ∈ B Z ∂ Ω φu φ d H N − , where u D = u χ D . Proof.
Let I = sup φ ∈ B Z ∂ Ω φu φ d H N − . Arguing as in the first part of the proof for Theorem 3.1 we have that I is finite.Next, let { φ i } i ≥ be a maximizing sequence and let u i = u φ i . It is clear that { u i } i ≥ is bounded in W ,p (Ω) , then there exists a function u ∈ W ,p (Ω) such that,for a subsequence that we still call { u i } i ≥ u i ⇀ u weakly in W ,p (Ω) ,u i → u strongly in L p (Ω) ,u i → u strongly in L p ( ∂ Ω) . On the other hand, since { φ i } i ≥ is bounded in L ∞ ( ∂ Ω), we may choose a subse-quence, again denoted { φ i } i ≥ , and φ ∈ L ∞ ( ∂ Ω) and such that φ i ∗ ⇀ φ weakly* in L ∞ ( ∂ Ω) . L. DEL PEZZO AND J. FERN´ANDEZ BONDER
Then I = lim i →∞ Z ∂ Ω φ i u i d H N − = 1 p − i →∞ n p Z ∂ Ω φ i u i d H N − − Z Ω |∇ u i | p + | u i | p d H N o ≤ p − n p Z ∂ Ω φu d H N − − Z Ω |∇ u | p + | u | p d H N o . Furthermore, by Theorem 2.3, there exists D ⊂ ∂ Ω with H N − ( D ) = A such that Z ∂ Ω φu d H N − ≤ Z ∂ Ω χ D u d H N − , and { t < u } ⊂ D ⊂ { t ≤ u } , t := inf { s : H N − ( { s < u } ) < A } . Thus I ≤ p − n p Z ∂ Ω χ D u d H N − − Z Ω |∇ u | p + | u | p d H N o . As a consequence of (3.2), we have that I ≤ p − n p Z ∂ Ω χ D u d H N − − Z Ω |∇ u | p + | u | p d H N o ≤ pp − n p Z ∂ Ω χ D u D d H N − − Z Ω |∇ u D | p + | u D | p d H N o = Z ∂ Ω χ D u D d H N − ≤ I. Recall that u D = u χ D . Therefore χ D is a solution to (5.1). This completes theproof. (cid:3) Remark 5.2.
Note that in arguments in the proof of Theorem 5.1, using again theTheorem 2.3, we can prove that { t < u D } ⊂ D ⊂ { t ≤ u D } where t := inf { s : H N − ( { s < u D } ) < A } . Therefore u D is constant on ∂D. Domain Derivative.
In this subsection we compute the shape derivative ofthe functional J ( χ D ) with respect to perturbations on the set D . We will considerregular perturbations and assume that the set D is a smooth subset of ∂ Ω.Then, by using the formula for the shape derivative, we deduce some necessaryconditions on a (regular) set D in order for it to be optimal for J in the L ∞ setting.Also, this formula could be used to derive algorithms in order to compute theactual optimal set (cf. with [10]).For the computation of the shape derivative, we use some ideas from [13].We begin by describing the kind of variations that we are considering on the set D . Let V be a regular (smooth) vector field, globally Lipschitz, with support in a ONLINEAR ELASTIC MEMBRANES 9 neighborhood of ∂ Ω such that h V, ν i = 0 and let ψ t : R N → R N be defined as theunique solution to(5.3) ( dd t ψ t ( x ) = V ( ψ t ( x )) t > ,ψ ( x ) = x x ∈ R N . We have ψ t ( x ) = x + tV ( x ) + o ( t ) ∀ x ∈ R N . Now, if D ⊂ ∂ Ω, we define D t := ψ t ( D ) ⊂ ∂ Ω.First, we compute the derivative at t = 0 of the surface measure of the set D t .That is, we want to compute dd t H N − ( D t ) (cid:12)(cid:12)(cid:12) t =0 . Lemma 5.3.
With the previous notation, if D ⊂ ∂ Ω is a smooth (relatively) openset, then dd t H N − ( D t ) (cid:12)(cid:12)(cid:12) t =0 = Z D divV d H N − . Proof.
We will use the following asymptotic formulae, for which the proofs can befound in [14]: Jψ t ( x ) = 1 + t divV(x) + o(t) , (5.4) [ ψ − t ] ′ ( x ) = Id − tV ( x ) + o ( t ) . (5.5)Then we have, by the change of variable formula, Proposition 2.5, H N − ( D t ) = Z D t d H N − = Z D | [ ψ − t ] ′ ( x ) ν | Jψ t ( x ) d H N − . Hence by (5.4), (5.5) and the definition of J τ we get, using that h V, ν i = 0, H N − ( D t ) = H N − ( D ) + t Z D div V d H N − + o ( t ) . Therefore, we arrive atdd t H N − ( D t ) (cid:12)(cid:12)(cid:12) t =0 = Z D div V d H N − . This is what we wanted to show. (cid:3)
Now, let I ( t ) = Z ∂ Ω u t χ D t d H N − , where u t ∈ W ,p (Ω) is the unique solution to(5.6) ( − ∆ p u t + | u t | p − u t = 0 in Ω , |∇ u t | p − ∂u t ∂ν = χ D t on ∂ Ωand assume that D ⊂ ∂ Ω is again a smooth (relatively) open set.We have the following Lemma:
Lemma 5.4.
Let u and u t be the solution of (5.6) with t = 0 and t > , respec-tively. Then u t → u in W ,p (Ω) , as t → + . Proof.
The proof follows exactly as the one in Lemma 4.2 in [6]. The only differencebeing that we use the trace inequality instead of the Poincar´e inequality. (cid:3)
Remark 5.5.
It is easy to see that, as ψ t → Id in the C topology, then fromLemma 5.4 it follows that w t := u t ◦ ψ t → u strongly in W ,p (Ω) . Now, we arrive at the main result of the section.
Theorem 5.6.
With the previous notation, if D ⊂ ∂ Ω is a smooth (relatively) openset, we have that I ( t ) is differentiable at t = 0 and dd t I ( t ) (cid:12)(cid:12)(cid:12) t =0 = pp − Z ∂D u h V, ν τ i d H N − , where u is the solution of (5.6) with t = 0 and ν τ stands for the exterior unitnormal vector to D along ∂ Ω . Proof.
By (3.2) we have that I ( t ) = sup v ∈ W ,p (Ω) p − (cid:26) p Z ∂ Ω vχ D t d H N − − Z Ω |∇ v | p + | v | p d H N (cid:27) . Given v ∈ W ,p (Ω) we consider u = v ◦ ψ t ∈ W ,p (Ω) , then, by the change ofvariables formula, Proposition 2.5, Z ∂ Ω vχ D t d H N − = Z ∂ Ω uχ D J τ ψ t d H N − = Z ∂ Ω uχ D d H N − + t Z ∂ Ω uχ D div τ V d H N − + o ( t ) . Also, by the usual change of variables formula, we have Z Ω |∇ v | p d H N = Z Ω | T [ ψ ′ t ] − ( x ) ∇ u T | p Jψ t d H N = Z Ω | ( I − t T V ′ + o ( t )) ∇ u T | p { t div V + o ( t ) } d H N = Z Ω {|∇ u | p − tp |∇ u | p − h∇ u, T V ′ ∇ u T i + o ( t ) }{ t div V + o ( t ) } d H N = Z Ω |∇ u | p d H N + t Z Ω |∇ u | p div V d H N − tp Z Ω |∇ u | p − h∇ u, T V ′ ∇ u T i d H N + o ( t ) , and Z Ω | v | p d H N = Z Ω | u | p Jψ t d H N = Z Ω | u | p d H N + t Z Ω | u | p div V d H N + o ( t ) . ONLINEAR ELASTIC MEMBRANES 11
Then, for all v ∈ W ,p (Ω) we have that p Z ∂ Ω vχ D t d H N − − Z Ω |∇ v | p + | v | p d H N = p Z ∂ Ω uχ D d H N − − Z Ω |∇ u | p + | u | p d H N + t (cid:20) p Z ∂ Ω uχ D div τ V d H N − Z Ω ( |∇ u | p + | u | p )div V d H N + p Z Ω |∇ u | p − h∇ u, T V ′ ∇ u T i d H N (cid:21) + o ( t ) . Therefore, we can rewrite I ( t ) as I ( t ) = sup u ∈ W ,p (Ω) p − { ϕ ( u ) + tφ ( u ) + o ( t ) } , where ϕ ( u ) = p Z ∂ Ω uχ D d H N − − Z Ω |∇ u | p + | u | p d H N and φ ( u ) = p Z ∂ Ω uχ D div τ V d H N − − Z Ω ( |∇ u | p + | u | p )div V d H N + p Z Ω |∇ u | p − h∇ u, T V ′ ∇ u T i d H N . If we define w t = u t ◦ ψ t for all t we have that w = u and I ( t ) = 1 p − { ϕ ( w t ) + tφ ( w t ) + o ( t ) } for all t . Thus I ( t ) − I (0) ≥ p − { ϕ ( u ) + tφ ( u ) + o ( t ) } − p − ϕ ( u ) , then(5.7) lim inf t → + I ( t ) − I (0) t ≥ p − φ ( u ) . On the other hand I ( t ) − I (0) ≤ p − { ϕ ( w t ) + tφ ( w t ) + o ( t ) } − p − ϕ ( w t ) , hence, I ( t ) − I (0) t ≤ p − φ ( w t ) + 1 t o ( t ) . By Remark 5.5, φ ( w t ) → φ ( u ) as t → + , therefore,(5.8) lim sup t → + I ( t ) − I (0) t ≤ p − φ ( u ) . From (5.7) and (5.8) we deduced that there exists I ′ (0) and I ′ (0) = 1 p − φ ( u )= 1 p − (cid:26) p Z ∂ Ω u χ D div τ V d H N − + p Z Ω |∇ u | p − h∇ u , T V ′ ∇ u T i d H N − Z Ω ( |∇ u | p + | u | p )div V d H N (cid:27) . Now we try to find a more explicit formula for I ′ (0).In the course of the computations, we require the solution u to ( − ∆ u + | u | p − u = 0 in Ω , |∇ u | p − ∂u ∂ν = χ D on ∂ Ω , to be C . However, this is not true. As it is well known (see, for instance, [19]), u belongs to the class C ,δ for some 0 < δ < ( − div(( |∇ u ε | + ε ) ( p − / ∇ u ε ) + | u ε | p − u ε = 0 in Ω , ( |∇ u ε | + ε ) ( p − / ∂u ε ∂ν = χ D on ∂ Ω . It is well known that the solution u ε to (5.9) is of class C ,ρ for some 0 < ρ < u ε and pass tothe limit as ε →
0+ at the end.We have chosen to work formally with the function u in order to make ourarguments more transparent and leave the details to the reader. For a similarapproach, see [13].Now, since div( | u | p V ) = p | u | p − u h∇ u , V i + | u | p div V, div( |∇ u | p V ) = p |∇ u | p − h∇ u D u , V i + |∇ u | p div V, we obtain I ′ (0) = 1 p − (cid:26) p Z ∂ Ω u χ D div τ V d H N − + p Z Ω |∇ u | p − h∇ u , T V ′ ∇ u T i d H N − Z Ω div(( |∇ u | p + | u | p ) V ) d H N + p Z Ω |∇ u | p − h∇ u D u , V i d H N + p Z Ω | u | p − u h∇ u , V i d H N (cid:27) . ONLINEAR ELASTIC MEMBRANES 13
Hence, using that h V, ν i = 0 in the right hand side of the above equality we find I ′ (0) = pp − (cid:26) Z ∂ Ω u χ D div τ V d H N − + Z Ω |∇ u | p − h∇ u , T V ′ ∇ u T + D u V T i d H N + Z Ω | u | p − u h∇ u , V i d H N (cid:27) = pp − (cid:26) Z ∂ Ω u χ D div τ V d H N − + Z Ω |∇ u | p − h∇ u , ∇ ( h∇ u , V i ) i d H N + Z Ω | u | p − u h∇ u , V i d H N (cid:27) . Since u is a week solution of (5.6) with t = 0 we have I ′ (0) = pp − (cid:26) Z ∂ Ω u χ D div τ V d H N − + Z ∂ Ω h∇ u , V i χ D d H N − (cid:27) = pp − Z ∂ Ω div τ ( u V ) χ D d H N − = pp − Z ∂D u h V, ν τ i d H N − . This completes the proof. (cid:3)
The following corollary is a result that we have already observed, actually underweaker assumptions on D, in Remark 5.2.Nevertheless, we have chosen to include this remark as a direct application ofthe Lemma 5.3 and Theorem 5.6.
Corollary 5.7.
Let χ D be a maximizer for J over the class B and assume that D ⊂ ∂ Ω is a smooth (relatively) open set. Let u D be the solution to the associatedstate equation ( − ∆ p u + | u | p − u = 0 in Ω , |∇ u | p − ∂u∂ν = χ D on ∂ Ω . Then, u D is constant along ∂D .Proof. Recalling the formula for the derivative of the volume, that is,dd t H N − ( D t ) (cid:12)(cid:12)(cid:12) t =0 = Z D div τ V d H N − = Z ∂D h V, ν τ i d H N − , and the fact that D is a critical point of I, we derive I ′ (0) = c dd t H N − ( D t ) (cid:12)(cid:12)(cid:12) t =0 ⇐⇒ u = constant, on ∂D. As we wanted to prove. (cid:3)
Final comments.
It would be interesting to say more about optimal config-urations. For instance: • What is the topology of optimal sets? Are optimal sets connected? • What about the regularity of optimal sets? Is it true that the boundary ofoptimal sets are regular surfaces? • Where are the optimal sets located?These questions, we believe, are difficult ones and we can only give an answerin the trivial case where the domain Ω is a ball. In this case, by symmetrizationarguments (by means of the spherical symmetrization , cf. with [12, 18]) it is straightforward to check that optimal sets are spherical caps.This example also shows that the uniqueness problem is far from obvious.
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Leandro M. Del PezzoDepartamento de Matem´atica, FCEyN, Universidad de Buenos Aires,Pabell´on I, Ciudad Universitaria (1428), Buenos Aires, Argentina.
E-mail address : [email protected] Juli´an Fern´andez BonderDepartamento de Matem´atica, FCEyN, Universidad de Buenos Aires,Pabell´on I, Ciudad Universitaria (1428), Buenos Aires, Argentina.
E-mail address : [email protected] Web page: http://mate.dm.uba.ar/ ∼∼