A MONOTONICITY RESULT FOR THE FIRST STEKLOV-DIRICHLET LAPLACIAN EIGENVALUE

. In this paper, we consider the ﬁrst Steklov-Dirichlet eigenvalue of the Laplace operator in annular domains with a spherical hole. We prove a monotonicity result with respect to the hole when the outer region is centrally symmetric. MSC 2020: Primary: 35J25, 35P15; Secondary: 28A75, 49Q10.


Introduction and main result
In the last years, the study of eigenvalue problems in perforated domains has been the object of much interest. These kind of problems are usually defined in annular domains with an outer region satisfying suitable assumptions and with a hole having spherical shape. In this case, different boundary conditions can be imposed on the outer and inner boundary of the domain and then several optimization problems can be studied (for instance Robin-Neumann [PPT], Neumann-Robin [DP], Dirichlet-Neumann [AA, AAK], Steklov-Dirichlet [PPS, GPPS, HLS], Steklov-Robin [GS]).
In this paper, we consider an eigenvalue problem for the Laplace operator in a suitable annular domain with both Steklov and Dirichlet boundary conditions. More precisely, let Ω 0 Ă R n , n ě 2, be an open, convex, bounded set with Lipschitz boundary and, let B r pyq be the ball of radius r ą 0 centered at y and such that B r pyq Ă 8 Ω 0 . We consider the following Steklov-Dirichlet eigenvalue problem where Ω is the annular domain Ω " Ω 0 zB r pyq, σ P R and ν is the outer unit normal to BΩ 0 .
The first eigenvalue of (1.1) has the following characterization σpΩq " min vPH 1 BBr pyq pΩq vı0 "ż Ω |Dv| 2 dx, }v} L 2 pBΩ 0 q " 1 where H 1 BBrpyq pΩq denotes the set of Sobolev functions which vanish on the boundary of B r pyq (see Section 2 for the precise definition).
In [PPS] (see also [D, HLS]), the authors prove that the minimum in (1.2) is achieved by a function u P H 1 BBr pyq pΩq which is a weak solution to problem (1.1) with constant sign in Ω, and that σpΩq is simple. Furthermore, the authors prove that, keeping the measure of Ω and the inner radius r fixed, among quasi-spherical sets, σpΩq is maximum when Ω is a spherical shell, that is when Ω 0 is a ball with the same center of the hole. On the other hand, in [GPPS] the authors extend this result to a class of annular sets with a suitable convex outer domain Ω 0 .
More properties are known when Ω is an eccentric spherical shell, that is, when the outer domain Ω 0 is a ball not necessarly centered at the same point of the hole. In [VS], the authors study the optimal placement of the hole in eccentric spherical shell Ω so that σpΩq is maximal keeping the outer ball and the inner radius fixed. They prove that σpΩq achieves the maximum when the two balls are concentric for n ě 3. Subsequently, this result has been also proved for any dimension n ě 2 in [F], by using different proofs (see also [S] for two-points homogeneous space).
Moreover, by performing numerical experiments, the authors in [HLS] exhibit that σpΩq is monotone decreasing with respect the distance between the centers of the two disks. Our aim is to prove that this monotonicity property holds in any dimension and in a more general setting.
Through this paper we will assume that the outer domain Ω 0 verifies the following assumption.
Definition 1.1. Let Ω 0 Ă R n , n ě 2, be an open, convex, bounded set with Lipschitz boundary and centrally symmetric with respect to x 0 P 8 Ω 0 , that is there exists x 1 P Ω 0 such that 1 2 px`x 1 q " x 0 for any x P Ω 0 . Let r ą 0 be fixed, our first question is pQ 1 q Where we have to place the center of the spherical hole with fixed radius r in order to maximize σpΩq? To give an answer, let w P R n be a unit vector and let us consider the holes B r ptq with the centers on the w-direction: where ρ w px 0 q " suptt ą 0 : x 0`t w P Ωu.
(1.4) We stress that ρ w is the distance between x 0 and the intersection point of the straight line on the direction w passing at x 0 and at the boundary of Ω 0 . Then we consider the following type of annular domains Ωptq " Ω 0 zB r ptq, (1.5) where Ω 0 verifies Definition 1.1. We stress that when Ω 0 is a ball, then the sets Ωptq are eccentric spherical shell. urthermore, for any 0 ď t ă ρ w px 0 q´r, we denote by σptq the first Steklov-Dirichlet eigenvalue of the Laplacian in Ωptq.
Our second questions is pQ 2 q Is σptq decreasing with respect to t? Our main result gives an answer to both questions pQ 1 q and pQ 2 q and it is the following.
Theorem 1.2. Let Ω 0 be as in Definition 1.1 and let B r ptq and Ωptq be defined as in (1.3) and (1.5), respectively. Let σptq be the first Steklov-Dirichlet eigenvalue of Ωptq, that is Then σptq is strictly monotone decreasing with respect to t.
As an immediate consequence of our main result, we obtain that, in order to maximize σptq, the hole has to be centered at the symmetry point of Ω 0 , when the inner radius is fixed.
To prove Theorem 1.2, we use a shape derivative approach. In particular we compute the first and the second domain derivatives of the first Steklov-Dirichlet eigenvalue. We emphasize that our result implies that the monotonicity property holds also for eccentric spherical shell in any dimension and in particular in two dimension as suggested by the numerical computation contained in [HLS].
Finally we describe the outline of the paper. In Section 2 we summarize some known results about the first Steklov-Dirichlet eigenvalue of the Laplace operator. Moreover we recall the definition of domain derivative and the Hadamard's formula. In the third section we compute the first shape derivative of σptq, observing that a stationary set occurs when the center of the hole coincides with the center of symmetry of the outer region. Finally, in the last section we compute the second shape derivative of σptq and prove our main result.

Background and preliminary results
2.1. The Steklov-Dirichlet Laplacian eigenvalue problem. Let Ω 0 Ă R n be an open, bounded, convex set with Lipschitz boundary and such that B r pyq Ă 8 Ω 0 , where B r pyq is the ball of radius r ą 0 centered at y. Let us consider the annular domain Ω :" Ω 0 zB r pyq. In what follows we denote the set of Sobolev functions on Ω vanishing on BB r pyq by H 1 BBrpyq pΩq, that is (see [ET]) the closure in H 1 pΩq of the following set C 8 BBrpyq pΩq :" tu| Ω | u P C 8 0 pR n q, sptpuq X BB r pyq " Hu. Let us consider the following Steklov-Dirichlet eigenvalue problem in Ω for the Laplace operator where ν is the outer unit normal to BΩ 0 . It is known (see for instance [A, D, P]), that the spectrum of (1.1) is discrete and the sequence of eigenvalues can be ordered as follows In particular, σpΩq, the first eigenvalue of (2.1), has the following variational characterization (see for instance [D, PPS, HLS]) Moreover in [PPS] (see also [HLS]), the authors prove the following result.
Proposition 2.1. Let r ą 0 and Ω 0 Ă R n be an open bounded connected set with Lipschitz boundary and such that B r pyq Ă 8 Ω 0 and let Ω :" Ω 0 zB r pyq. There exists a function u P H 1 BBr pyq pΩq which achieves the minimum in (2.2) and it is a weak solution to the problem (2.1). Moreover σpΩq is simple and the first eigenfunctions have constant sign in Ω.

2.2.
Domain derivative and Hadamard's formula. In this Section, we briefly recall the Hadamard formulas in the framework of the domain derivative (see for instance [SZ, B, HP]).
Let E Ă R n be an open bounded set with Lipschitz boundary and let V pxq a vector field such that V P W 1,8 pR n ; R n q. For any t ą 0, let Eptq " txptq " x`tV pxq, x P Eu, and f pt, xptqq be such that f pt,¨q P W 1,1 pR n q and it is differentiable at t. Then where ν is the outer unit normal to the boundary of Eptq and H n´1 denotes the Hausdorff measure (see for instance the Chapter 5 of [HP]).
Moreover, when the integral is defined on the oriented boundary of E, if f pt,¨q P W 2,1 pR n q, it holds: where H denotes the mean curvature of BEptq (see for instance Chapter 5 of [HP]).

First shape derivative of σptq
Let Ω 0 be as in Definition 1.1. From now on for the reader convenience, we assume that x 0 " 0. In the sequel we denote by B r and by ρ, the ball centered at the origin with radius r and ρp0q, respectively, and we set Ω " Ω 0 zB r (see the figure below).
BΩ0 Ω " Ω0zBr r Br In order to compute the first and the second domain derivative, let us define a suitable smooth vector field which move the hole B r in a given direction w keeping the boundary of Ω 0 and the inner radius fixed and hence with perturbed holes having the form in (1.3). We consider the following variational field in R n : where ϕ P C 8 0 pΩ 0 q is a cut-off function such that ϕpxq " 1 on B r . Therefore the perturbed annular domains Ωptq, defined in (1.5), can be seen as where ρ w is defined in (1.4), see also the figures. We observe that BΩptq " BΩ 0 Y BB r ptq and that Ωptq is centrally symmetric if and only if t " 0 and the center of symmetry is the origin.
For any 0 ď t ă ρ w´r , let σptq be the first Steklov-Dirichlet eigenvalue of the Laplacian in Ωptq and u t be the corresponding normalized and positive eigenfunction.
and it is a solution to the following problem In the sequel ν will denotes the outer unit normal to the boundary of the annular domain BΩptq. By the variational characterization of σptq we have σptq " ż Ωt |∇u t | 2 dx.
We observe that by standard arguments on shape derivatives (see for instance [HP]), it follows that u t and σptq are differentiable with respect to t. For the sake of completeness, we give here sketchily the proof that follows exactly the same arguments contained in [HP]. The main tool is a general version of the implicit function theorem (see [HP] and also [Bo,Lem.2.1] and [Sc]) applied to the equation transferred onto the fixed domain Ω 0 (we refer for the details to [HP] and also to [Bo,Lem. 2

.7] and [HLS, Th.1]).
Proposition 3.1. Let σptq and u t be respectively the first eigenvalue to the problem (3.3) and the corresponding normalized eigenfunction and let ρ w be as in (1.4). Then the functions t P p0, ρ w´r q Ñ σptq, t P p0, ρ w´r q Ñ u t are differentiable for any direction w P S n´1 and for any t P p0, ρ w´r q.
Proof. Let us fix t P r0, ρ w´r r and let s ą 0 be such that t`s ă ρ w´r and let us consider σpt`sq and u t`s . By definition, the following weak formulation holds: ż Let V be as in (3.1) and let us define the following map Φ : ps, xq P p´ρ w`r´t , ρ w´r´t qˆΩ 0 Ñ x`sV pxq P R n .
Now, let us denote by pH 1 BBr ptq pΩqq 1 the dual space of H 1 BBrptq pΩq and let us define f : ps, v, σq P RˆH 1 BBrptq pΩqˆR Ñ pf 1 , f 2 q P pH 1 BBrptq pΩqq 1ˆR , where # xf 1 ps, v, σq, Ψy " ş Ω 0 pp∇vqpDΦps,¨qq´1qpp∇ΨqpDΦps,¨qq´1q|DΨps,¨q|dx´σ BBr ptq pΩ 0 q. If we consider the function g : s P U Ñ`u t`s pΦps,¨q, σpt`sq˘P H 1 BBrptq pΩqˆR, then f ps, gpsqq " 0 for any s P U and gp0q " pu t , σptqq. In order to obtain the claim we have to prove that g is differentiable in s " 0, that follows by applying the implicit function Theorem. To do this, we have to prove that Bf Bpv, σqˇˇˇ0,u t ,σptq : is an isomorphism. This can be proved following line by line the same arguments of [Bo,Lem 2.7] and [HLS,Th. 1] (see also chapter 5 in [HP]).
Remark 3.2. We observe that being u t harmonic for any t ě 0, then u t P C 8´Ω ptq¯. Then by the general theory of the shape derivatives (see Chapter 5 of [HP]), follows that u t is C 8 in a neighborhood of t.
For sake of simplicity, in what follows, we use u in place of u t . By (1.3) we have d dt upt, xptqq " u 1`x ∇u, wy, (3.5) where u 1 " Bu Bt . Recalling that the perturbed hole B r ptq is the zero-level set of the function u, then (3.5) gives for any 0 ď t ă ρ w´r . Since u is a solution to (3.3), by (3.6) we get that u 1 is a solution to the following problem Before to prove our main result, we finally observe that, if we derive the normalized condition (3.2) by using the Hadamard's formula (2.3), we get Now we can compute the first derivative of σptq. where u the is the normalized, positive eigenfunction corresponding to σptq.
Proof. By using Hadamard's formula, since u solves (3.3), and observing that the unit outer normals νpxq to BB r and νpxptqq to BB r ptq coincide, we get BBrptq |∇u| 2 xν, wy dH n´1 .
The symmetry of Ω 0 implies the following.
Corollary 3.4. In the same assumptions of Theorem 3.3, it holds d dt σp0q " 0.
Proof. Let u be the normalized, positive eigenfunctions corresponding to σp0q " σpΩq. We observe that since Ω 0 is centrally symmetric with respect the origin (see Definition 1.1), then x 1 "´x and it holds upxq " up´xq.
This follows immediately by taking vpxq " upx 1 q as test function in (1.2) and being the eigenfunction u normalized. Then by Theorem 3.3 and being ν "´x r , we have d dt σp0q "´ż BBr |∇u| 2 xν, wy dH n´1 " 0.

Second shape derivative of σptq
In this Section, before to prove our main result (Theorem 1.2), we need to compute the second domain derivative of σptq. We use the same notations introduced to compute the first shape derivative.
Proof. In order to compute the second derivative we first observe that
The Theorem 1.2 follows by applying the previous result.
Proof of the Theorem 1.2. The claim is a direct consequence of the Theorems 3.3, Corollary 3.4 and Theorem 4.1. We stress that the second derivative in (4.1) cannot be zero by the convexity of Ω 0 and the Hopf Lemma.
Remark 4.2. An immediate consequence of Theorem 1.2 is that σptq is maximum when the hole has to be centered at the symmetry point of Ω 0 , as the inner radius is fixed.
Remark 4.3. We stress that in [HLS] the authors prove the following estimate in two dimension when Ω 0 " B R , with R ą r lim inf tÑpR´rq´σ ptq ě r 2RpR´rq , (4.9) where σptq " σpΩptqq and we recall that Ωptq " B R zB r ptq. By our Theorem 1.2 the estimate (4.9) can be written as the following lower bound 1 R logp R r q " σpB R zB r q ě σptq ě r 2RpR´rq , @t Ps0, R´rr. (4.10) We stress that inequality (4.10) gives an upper and lower bound for σptq in terms of the two radius of the eccentric annulus.