Shape optimization for an elliptic operator with infinitely many positive and negative eigenvalues

The paper deals with an eigenvalue problems possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domain.


Introduction
In this paper we study the spectrum of the problem ∆u + λu = 0 in Ω ∂ ν u = λ σ u in ∂Ω, (1.1) where σ, λ ∈ R, and Ω ⊂ R n is a bounded domain with smooth boundary. The corresponding Rayleigh quotient is For positive σ the Rayleigh quotient is positive and the classical theory for symmetric operators applies. François [14] has shown hat in this case the spectrum consists of countably many eigenvalues, which are bounded from below and tend to infinity.
Our interest in this paper is directed to the problem with σ negative. It has been studied in [6] and [4]. A more general approach is found in [10]. It is known that in addition to λ = 0 there exist two sequences of eigenvalues, one tending to +∞ and the other tending to −∞. The eigenfunctions are complete in H 1 (Ω) except in the resonance case |Ω| + σ|∂Ω| = 0 where some supplements are required (see [6]).
The smallest positive eigenvalue λ 1 (Ω) and the largest negative eigenvalue λ −1 (Ω) play the role of fundamental frequencies. Based on the isoperimetric inequalities for the fundamental frequency of the membrane we study the dependence of λ ±1 (Ω) on some geometrical properties such as the volume and the harmonic radius.
We first establish inequalities by means of the harmonic transplantation which is appropriate for this type of problems. An interesting question is whether the Rayleigh-Faber -Kahn inequality extends to these eigenvalues. Here only answers for nearly spherical domains can be provided. The arguments are based on the first and second order shape derivatives. For general domains the answer is still incomplete.
One motivation for studying this problem are dynamical boundary conditions for parabolic equations. A simple version is given by the heat equation in Ω.
Such problem are well studied (see e.g. [11] - [13], [15], [19], [20]). It is known that they are well posed for positive σ in the space C(0, T, H 1,2 (Ω)), in the sense of Hadamard, and that there exists a smooth solution globally in time, whereas this is not the case if σ < 0. That is, there is no continuous dependence on the initial conditions (except in dimension one).
The paper is organized as follows. First we present the eigenvalue problem and quote some known results. Then we derive inequalities by means of the method of harmonic transplantation. In the last part we compute the first and the second domain variations of the fundamental eigenvalues and derive some inequalities and monotonicity properties of nearly circular domains.

The eigenvalue problem, known results
Let Ω ⊂ R n be a bounded domain with smooth boundary. Assume σ > 0. In that case (1.2) gives a non-negative functional R σ on the Hilbert space H 1,2 (Ω). Its minimum is equal to zero and is achieved by any constant function. If we minimize R σ over the set the direct method of the calculus of variations gives us a unique minimizer which is non-negative and solves (1.1). By Harnack's inequality it is positive in Ω. G. François in [14] showed that there exist countably many eigenvalues (λ n ) n which tend to ∞, for which (1.1) admits solutions.
The case σ < 0 was considered in [3] and [6] and will be under consideration from now on. For u, v ∈ H 1,2 (Ω) let be an inner product on L 2 (Ω) ⊕ L 2 (∂Ω). We define for all u ∈ K.
In [4] and [6] the authors showed the existence of two infinite sequences of eigenvalues. One sequence consists of negative eigenvalues (λ −n ) n and the other of positive eigenvalues (λ n ) n . The corresponding eigenfunctions (u ±n ) n ∈ K solve (1.1). Moreover The eigenvalues are ordered as Note that λ ±n = λ ±n (σ) for n ∈ N.
In [6] it was also shown, that the quantity plays an important role. If σ = σ 0 the following characterization of λ ±1 holds : For n = 1 (and σ = σ 0 ) the eigenvalues λ ±n have a variational characterization as well. Indeed, let λ ±i , i = 1, . . . , k be the first k eigenvalues, counted with their multiplicities. Let u i be the corresponding eigenfunctions. Then we define We get the characterization Remark 1 It is interesting to note, that in the case σ = σ 0 the constraint a(u ±1 , 1) = 0 is satisfied automatically. This can be seen by the following considerations. Let u ±1 be an eigenfunction corresponding to λ ±1 . Then necessarily for all v ∈ H 1,2 (Ω). The real numbers µ ±1 andμ ±1 denote the Lagrange parameters for the constraints. The special choice v = u ±1 ∈ K gives For v = 1 we getμ ±1 (|Ω| + σ|∂Ω|) = 0.
By our assumptions on σ we get In the same way we prove In [6] (Theorem 14, Theorem 21 and Corollary 22) the following result was proved.
Proof We assume σ 1 > σ 2 . We distinguish two cases. Case 1. From the characterization of λ 1 we get For u we choose the eigenfunction of λ 1 (σ 2 ) and we obtain .
Remark 3 In [6] the authors also studied the smoothness and asymptotic behaviour of the map They proved that is a smooth curve with the following asymptotics: where µ D is the first Dirichlet eigenvalue for the Laplacian.
We are interested in the domain dependence of λ ±1 . Thus we will write λ 1 = λ 1 (Ω) and λ −1 = λ −1 (Ω). Note that σ 0 = σ 0 (Ω) depends on Ω as well. Moreover for domains of given volume and for a ball B R with the same volume the isoperimetric inequality gives In [3] and [6] the following properties were proved.
Lemma 3 For some given ball B R and σ 0 (B R ) = − R n let λ 1 and λ −1 be given by (2.1). Then the following cases occur.

Harmonic transplantation
From Section 2 we know that the eigenvalues λ 1 (resp. λ −1 ) have a variational characterization for σ = σ 0 (Ω). Moreover for σ < σ 0 (Ω) has a unique minimizer (of constant sign). The same holds in the case σ 0 (Ω) < σ < 0 for We are interested in optimality results for these eigenvalues. They will be obtained by means of the method of harmonic transplantation which was introduced by Hersch [17], (cf. also [2]). It generalizes the conformal transplantation used in complex function theory. In [9] it was applied to some shape optimization problems involving Robin eigenvalues. For convenience we shortly review some of the principal properties. For his mehod we need the Green's function with Dirichlet boundary condition For fixed y ∈ Ω the funcion H(·, y) is harmonic.

Definition 1
The harmonic radius at a point y ∈ Ω is given by The harmonic radius vanishes on the boundary ∂Ω and takes its maximum r Ω at the harmonic center y h . It satisfies the isoperimetric inequality ( [17], [2]) To illustrate the size of the harmonic radius we note that r Ω is estimated from below by the inner radius r i (Ω) and from above by the outer radius r o (Ω) of the domain: Note that G B R (x, 0) is a monotone function in r = |x|. Consider any radial function φ : B r Ω → R thus φ(x) = φ(r). Then there exists a function ω : R → R such that To φ(x) we associate the transplanted function U : Ω → R defined by U (x) = ω(G Ω (x, y h )). Then for any positive function f (s), the following inequalities hold true For a proof see [17] or [2] and in particular [9] for a proof of (3.6). The following observation will be useful in the sequel.
Consider first the case λ σ 1 (Ω) with σ < σ 0 = |Ω| |∂Ω| . Let u be a positive normalized radial eigenfunction of Problem (1.1) in B r Ω with σ replaced by σ , corresponding to the eigenvalue λ σ 1 (B r Ω ). Here Taking into account (3.5) and Remark 5 we get By (3.7) the right-hand side is positive and is equal to 1/λ σ 1 (B r Ω ). Consequently Consider now the case σ 0 < σ < 0. Define Let u be a positive normalized radial eigenfunction of problem (1.1) in B r Ω corresponding to the eigenvalue λ σ 1 (B r Ω ). Let U be the transplanted function of u in Ω. Then by (2.3) we get We apply (3.6) to the first integral in the denominator and again Remark 5 to the second.
We may rewrite this inequality as This proves the following theorem.

Equality holds in both cases if and only if Ω is a ball.
Since σ ≤ σ, Lemma 2 gives In general σ and σ are not comparable.

Remark 6
It is interesting to compare 1) in Theorem 1 with (2.6) in Lemma 3 (i). We get the following two sided bounds. If B R is a ball of equal volume with Ω and if σ < σ 0 (B R ) < 0 then Equality holds for the ball.
4 Domain dependence

Small perturbations of a given domain
We are interested in deriving optimality conditions for for the domain functionals λ ±1 (Ω). Contrary to the results in [3] (see (2.6) and (2.7)) these results will be local. We first decribe the general setting.
Consider a family of domains (Ω t ) t . The parameter t varies in some open interval (−t 0 , t 0 ) where t 0 > 0 is prescribed. With this notation we set Ω 0 := Ω. The family is given by the following construction. Let be a smooth family, where v and w are vector fields such that v, w : Ω → R n are in C 1 (Ω).
Note that for t 0 > 0 small enough (Φ t ) |t|<t 0 is a family of diffeomorphism. This restricts t 0 and defines the notion of "small perturbation of Ω".
The volume of Ω t is given by where J(t) is the Jacobian determinant corresponding to the transformation Φ t . The Jacobian matrix corresponding to this transformation is up to second order terms of the form By Jacobi's formula we have for small t Here we used the notation For the first variation we have only to require that y is volume preserving of the first order, that is We also consider perturbations which, in addition to the condition (4.3), satisfy the volume preservation of the second order, namely In addition we consider perturbations which preserve the surface area up to second order. Instead of (4.2) we then use the expansion From this we can derive first and second order conditions. In [7] the following first order condition ∂Ω (v · ν)H ∂Ω dS = 0 (4.5) and second order condition were derived. Here F (∇ * v, v) is a known scalar function of the tangential derivative ∇ * v of v (see e.g. formula (2.20) in [7]). Moreover, H ∂Ω denotes the mean curvature of ∂Ω. In particular for the ball B R this condition reads as where the last term in the sum is computed in (4.4) (see also Lemma 2 in [7]). For later use we setS Note that for volume preserving perturbations,S(0)(=S 0 (0)) describes the isoperimetric defect and is strictly positive (see also Section 7 in [7]).

The first variation and monotonicity
Let (Ω t ) t be a smooth family of small perturbations of Ω as described in the previous subSection. In particular they will be either volume preserving in the sense of (4.3) and (4.4) or area preserving in the sense of (4.5) and (4.6). For the moment we denote by λ either of the two first eigenvalues λ ±1 . We denote by u t (x) := u(y(x); t) the solution of where u t solves (4.9). Consequently the energy is In [7] and more detailed in [8] the first and second variation of E with respect to t were computed. For the first variation we obtained (see (4.1)) in [7]) Remark 7 The differentiability of λ(t) in t = 0 is not automatic. In fact, the eigenvalues λ = λ ±1 are differentiable in t = 0 if λ ±1 is simple (see e.g. [18,IV,Sec. 3.5]). As we know from Lemma 1 (i) and (ii) in Section 2, this is true for our choice of σ.
The conditionλ(0) = 0 and (4.12) gives the necessary condition In the case of volume preserving perturbation, (4.3) implies This is a special case of Theorem 1 in [7]. In the case of surface area preserving perturbations we apply (4.5) and obtain as a necessary condition for any critical point of λ(Ω). It is an open question whether (4.13) or (4.14) implies, that Ω can only be a ball.
From now on let Ω = B R . To ensure differentiability of λ in t = 0 we consider the cases for Then (4.13) and (4.14) are satisfied since the corresponding eigenfunctions are radial. Henceλ(0) = 0, i.e. the ball is a critical domain. Formula (4.12) implies monotonicity of λ ±1 for nearly spherical domains with respect to volume increasing (decreasing) perturbations. Indeed we rewrite (4.12) as Then we use the boundary condition u 2 r (R) = λ 2 σ 2 u 2 (R) and obtain Next we determine the sign of k(R). In this we modify the proof of Lemma 3 in [9]. For the sake of completeness we give the details. Lemma 4 Let k(R) be given by (4.18) and let u(r) be the positive radial function in the case λ = λ 1 or λ = λ −1 . Then we have Proof In the radial case either eigenfunction satisfies the differentia equation We set z = ur u and observe that dz dr At the endpoint dz dr We know that z(0) = 0 and z(R) = λ σ. Note that We distinguish two cases.
We easily prove the following lemma.
Lemma 5 The first derivativeλ ±1 (0) satisfies the following sign conditioṅ Proof This follows directly from (4.17) and Lemma 4. Indeed it is sufficient to recall that a(u, u) > 0 for λ = λ 1 (B R ) and a(u, u) < 0 for λ = λ −1 (B R ). Also note that in either case u is positive.

The second variation
We are interested in extremality properties of the ball. We set u = ∂ t u t (x)| t=0 . This quantity is called "shape derivative" and plays a crucial role in determining the sign of the second domain variation of λ. In a first step we derive an equation for u . This follows the technique in [9]. Another good reference, with a slightly different approach, is also the book of A. Henrot an M. Pierre [16]. As a result we consider the following boundary value problem.
where k(R) is given in (4.18).
To (4.22) we associate the quadratic form We now turn to the computation ofλ(0). If we write (4.11) as If we useλ(0) = 0, we obtain the formulä In the case Ω = B R , we repeat computations as done in [7] (Section 7). However we don't assume any more, that the perturbations are volume preserving. This leads to the following modified formula (note that F 2 (0) = a(u, u)) In view of (2.1), (4.15) and (4.16) the ball B R is a local minimizer for λ if In the next section we will discuss the sign ofλ(0).

The sign of the second variation
We consider the following Steklov eigenvalue problem There exists an infinite number of eigenvalues Remark 8 Note that for λ = λ ±1 we have an eigenvalue µ = 0. Indeed, the case µ = 0 corresponds to the case where φ = u ±1 . For σ < σ 0 (B R ) (resp. σ 0 (B R ) < σ < 0) the eigenvalue λ 1 (resp. λ −1 ) is simple and the eigenfunction u 1 (resp. u −1 ) is of constant sign. Thus 0 = µ = µ 1 . As a consequence the spectrum consists only of non-negative eigenvalues.
There exists a complete system of eigenfunctions Similarly to the dicussion in Section 7 in [7] we get the representation for the solution u of (4.22) and the perturbation v · ν. Note that by (4.3) or (4.5) we have c 1 = b 1 = 0.
It is easy to check that where k(R) is defined in (4.18). We will also use the estimatë This was shown in the derivation of (7.13) in [7] and holds equally in the case (4.3) or (4.5).

Volume preserving perturbations
In this case we haveV (0) = 0. We first consider λ 1 . According to (4.24), a necessary condition for the ball to be a minimizer is Since λ 1 > 0 and σ < 0 we can apply (4.29), (4.27) and (4.28). Thus it is sufficient to show that where µ 2 is the first positive eigenvalue of the Steklov eigenvalue problem (4.26).
Such an expression also appeared in [7] Section 7.2.2 subsection 2 where the second domain variation of a Robin eigenvalue problem was considered. It was shown that If we set λ = λ 1 , α = −λ 1 σ and take into account (4.18) this is equivalent to This proves (4.30).
Theorem 2 For some given ball B R let σ 0 (B R ) = − R n . Let λ 1 be given by (2.1). Then the following optimality result holds: If σ < σ 0 (B R ), then among all smooth domains of equal volume the ball B R is a local minimizer for λ 1 . If we exclude translations and rotations of B R then it is a strict local minimizer.
The case λ = λ −1 (B R ) < 0 and σ 0 (R) < σ < 0 is similar. In that case k(R) < 0 by Lemma 4. SinceV (0) = 0, we deduce from (4.25) the following necessary condition for the ball to be a minimizer: Note that in this case λ −1 σ > 0, thus we can apply (4.29) again. As a consequence (4.31) holds, if In view of (4.27) and (4.28) this is equivalent to Note that the first term in the sum is positive, the second is positive as well, while the third term is negative. Consequently it suffices to show This is equivalent to we set σ := −δ R n for some 0 < δ < 1. Then (4.32) reads as n + 1 Inequality (4.33) is a quadratic inequality in λ −1 . It is easy to check that both zeros of the quadratic exprssion are negative for all 0 < δ ≤ 1. This proves the minimality of the ball for λ −1 for all σ 0 (B R ) < σ < 0.
Theorem 3 For some given ball B R let σ 0 (B R ) = − R n . Let λ −1 be given by (2.1). Then the following optimality result holds: If σ 0 (B R ) < σ < 0, then among all smooth domains of equal volume the ball B R is also a local minimizer for λ −1 . If we exclude translations and rotations of B R then it is a strict local minimizer.

Area preserving perturbations
In this caseS(0) and (4.8) gives We first consider λ 1 . According to (4.24) and (4.34), a necessary condition for the ball to be a minimizer is −2Q(u ) − λ 1 σu 2 (R)S 0 (0) + R n + 1 k(R)u(R)S 0 (0) (4.35) −2λσk(R)u(R) If we compare this with (4.30) we see that an extra positive term occurs. Since the arguments in [7] Section 7.2.2 subSection 2 carry over to the area preserving case without any changes we proved the following theorem.
Theorem 4 For some given ball B R let σ 0 (B R ) = − R n . Let λ 1 be given by (2.1). Then the following optimality result holds: If σ < σ 0 (B R ), then among all smooth domains of equal area the ball B R is a local minimizer for λ 1 . If we exclude translations and rotations of B R then it is a strict local minimizer.
Theorem 5 For some given ball B R let σ 0 (B R ) = − R n . Let λ −1 be given by (2.1). Then the following optimality result holds: If σ 0 (B R ) < σ < 0, then among all smooth domains of equal area the ball B R is also a local minimizer for λ −1 . If we exclude translations and rotations of B R then it is a strict local minimizer.
5 Open problems 1. The variational characterization of λ ±1 (see (2.1)) is also related to two inequalities, known as Friedrich's inequality and trace inequality. In fact, for σ < σ 0 < 0 we get Friedrich's inequality (5.1) and for σ 0 < σ < 0 we get The second inequality was also considered in [1] where also the case of equality was analyzed.
It is an interesting open problem to find explicit lower bounds for λ 1 (Ω) and σλ −1 (Ω). Note that the technique of harmonic transplantation gives upper bounds for these two quantities.
2. At least for λ −1 (Ω) it may be true that the ball of equal volume is only a local minimizer. There is no global result available at the moment. Therefore -motivated by Theorem 1 2) -it may also be interesting to ask if a quantity like |Ω|λ −1 (Ω) has the ball of equal volume as a minimizer.