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

Catherine Bandle 1  and Alfred Wagner 2
  • 1 Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051, Basel, Switzerland
  • 2 Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany
Catherine Bandle
  • Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051, Basel, Switzerland
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and Alfred Wagner

Abstract

This paper deals with an eigenvalue problem 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 domains.

1 Introduction

In this paper, we study the spectrum of the problem

{Δu+λu=0in Ω,νu=λσuon Ω,

where σ,λ and Ωn is a bounded domain with smooth boundary. The corresponding Rayleigh quotient is

Rσ(v)=Ω|v|2𝑑xΩv2𝑑x+σΩv2𝑑S,vH1,2(Ω).

For positive σ, the Rayleigh quotient is positive and the classical theory for symmetric operators applies. François [15] has shown that, in this case, the spectrum consists of countably many eigenvalues which are bounded from below and tend to infinity.

Our interest in this paper concerns the problem with σ negative. This case has been studied in [4, 6]. A more general approach is found in [11]. It is known that, in addition to λ=0, there exist two sequences of eigenvalues, one tending to + and the other to -. The eigenfunctions are complete in H1(Ω) except in the case of resonance|Ω|+σ|Ω|=0, where some supplements are required (see [4]).

The smallest positive eigenvalue λ1(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 geometric 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-order 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

{tu-Δu=0in (0,)×Ω,σtu+νu=0on (0,)×Ω,u(x,0)=u0(x)in Ω.

Such problems are well studied (see, for example, [12, 13, 14, 18, 20, 21]). It is known that they are well-posed for positive σ in the space C([0,T],H1,2(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 we derive some inequalities and monotonicity properties for nearly circular domains.

2 The eigenvalue problem and known results

Let Ωn be a bounded domain with smooth boundary. If σ>0, the Rayleigh quotient is non-negative. In this case, it was shown by François [15] that the eigenvalue problem (1.1) has countably many eigenvalues

0=λ0<λ1λ2λn

such that λk as k.

The case σ<0 was considered in [2, 4] and will be studied in the following. For u,vH1,2(Ω), let

a(u,v):=Ωuv𝑑x+σΩuv𝑑S

be an inner product on L2(Ω)L2(Ω). We define

𝒦:={uH1,2(Ω):Ω|u|2𝑑x=1,a(u,1)=0}.

Then, from (1.2) we have

Rσ(u)=1a(u,u)for all u𝒦.

In [6, 4], the authors showed the existence of two infinite sequences of eigenvalues. One sequence consists of negative eigenvalues (λ-k)k and the other of positive eigenvalues (λk)k. The corresponding eigenfunctions (u±k)k𝒦 solve (1.1). The eigenvalues are ordered as

λ-kλ-1<λ0=0<λ1λ2λk.

Note that λ±k=λ±k(σ) for k. If n>1, we have

limkλ-k=-andlimkλk=.

If n=1, there exist infinitely many positive eigenvalues, but only finitely many negative ones.

For k>1 (and σσ0) the eigenvalues λ±k have a variational characterization as well. Indeed, let λ±i, i=1,,k, be the first k eigenvalues, counted with their multiplicities. Let ui be the corresponding eigenfunctions. Then, we define

𝒦k:={uH1,2(Ω):Ω|u|2𝑑x=1,a(u,1)=0,a(u,ui)=0,i=1k}.

We get the characterization

λk+1(σ)=1supu𝒦ka(u,u)>0andλ-k-1(σ)=1infu𝒦ka(u,u)<0.

Remark 2.1.

Let λ1 be given by (2.1), that is,

1λ1(σ)=supu𝒦a(u,u).

Assume that σ<σ0 and let uH1,2(Ω) only satisfy

Ω|u|2𝑑x=1.

Then, u it is not admissible, since a(u,1) may be different from zero. Let c be chosen as

c:=-a(u,1)a(1,1).

With this choice we have u+c𝒦 and

a(u+c,u+c)=a(u,u)+2ca(u,1)+c2a(1,1)=a(u,u)-a(u,1)2a(1,1).

By our assumptions on σ we have a(1,1)<0 and, therefore,

1λ1(σ)a(u+c,u+c)a(u,u).

In the same way, we prove that

1λ-1(σ)a(u,u)

for 0>σ>σ0.

Lemma 2.2.

The eigenvalues λ1(σ) for σ<σ0 and λ-1(σ) for 0>σ>σ0 are monotone decreasing in σ.

Proof.

We assume that σ1>σ2. We distinguish two cases.

Case 1. From the characterization of λ1 we get

1λ1(σ1)Ωu2𝑑x+σ1Ωu2𝑑SΩu2𝑑x+σ2Ωu2𝑑S.

For u, we choose the eigenfunction of λ1(σ2) and we obtain

1λ1(σ1)1λ1(σ2).

This gives λ1(σ1)λ1(σ2).

Case 2. From the characterization of λ-1 we get

1λ-1(σ2)Ωu2𝑑x+σ2Ωu2𝑑SΩu2𝑑x+σ1Ωu2𝑑S.

In this case, we choose u as the eigenfunction of λ-1(σ1) and we obtain

1λ-1(σ2)1λ-1(σ1).

Since λ-1(σ)<0, we have λ-1(σ1)λ-1(σ2). ∎

Remark 2.3.

In [4], the authors also studied the smoothness and the asymptotic behavior of the map

σλ±1(σ).

They proved that

λ(σ)={λ1(σ)if σ<σ0,0if σ=σ0,λ-1(σ)if 0>σ>σ0,

is a smooth curve with asymptotics

limσ-λ(σ)=μDandlimσ0λ(σ)=-,

where μD is the first Dirichlet eigenvalue of the Laplacian.

We are interested in the domain dependence of λ±1. Thus, we will write λ1=λ1(Ω) and λ-1=λ-1(Ω). Note that σ0=σ0(Ω) depends on Ω as well. Moreover, for domains of given volume and for a ball BR with the same volume, the isoperimetric inequality gives

σ0(Ω)=-|Ω||Ω|=-|BR||Ω|-|BR||BR|=σ0(BR).

In [2], the following properties were proved.

Lemma 2.4.

Let BR be a ball with the same volume as Ω.

  1. (i)If σ<σ0(BR), then
    λ1(Ω)λ1(BR).
  2. (ii)For any domain Ω , there exists a number σ^(σ0(Ω),0) such that
    λ-1(Ω)λ-1(BR)
    whenever σ(σ0(Ω),σ^).

Remark 2.5.

For (i), we note that the condition σ<σ0(BR) is more restrictive than the condition σ<σ0(Ω) if |Ω|=|BR|. This is a consequence of (2.4).

3 Harmonic transplantation

From (2.2) and (2.3) it follows that

0<1λ1(Ω)=sup{Ωv2𝑑x-|σ|Ωv2𝑑S:Ω|v|2𝑑x=1}

for σ<σ0. Furthermore, there exists a unique positive maximizer. Similarly, for σ0(Ω)<σ<0, we have

0>1λ-1(Ω)=inf{Ωv2𝑑x-|σ|Ωv2𝑑S:Ω|v|2𝑑x=1}.

Also in this case, there exists a unique positive minimizer. It should be mentioned that the case σ=σ0 requires a different argument and cannot be treated with the arguments presented in this paper (cf. [4]).

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 in [17] (cf. also [3]). It generalizes the conformal transplantation used in complex function theory. In [7], it was applied to some shape optimization problems involving Robin eigenvalues. For convenience, we shortly review some of the principal properties. For this method, we need the Green’s function of the Laplacian with Dirichlet boundary conditions, that is,

GΩ(x,y)=γ(S(|x-y|)-H(x,y)),

where

γ={12πif n=2,1(n-2)|B1|if n>2,

and

S(t)={-lntif n=2,t2-nif n>2.

For fixed yΩ, the function H(,y) is harmonic.

Definition 3.1.

The harmonic radius at a point yΩ is given by

r(y)={e-H(y,y)if n=2,H(y,y)-1n-2if n>2.

The harmonic radius vanishes on the boundary Ω and takes its maximum rΩ at a harmonic center yh. It satisfies the isoperimetric inequality (see [17, 3])

|BrΩ||Ω|.

To illustrate the size of the harmonic radius, we note that rΩ is estimated from below by the inner radius ri(Ω) and from above by the outer radius ro(Ω) of the domain, that is,

ri(Ω)rΩro(Ω).

Note that GBR(x,0) is a monotone function in r=|x|. Consider any radial function ϕ:BrΩ such that ϕ(x)=ϕ(r). Then, there exists a function ω: such that

ϕ(x)=ω(GBrΩ(x,0)).

We associate to ϕ the transplanted function U:Ω defined by U(x)=ω(GΩ(x,yh)). Then, for any positive function f(s), we have the inequalities

Ω|U|2𝑑x=BrΩ|ϕ|2𝑑x,
Ωf(U)𝑑xBrΩf(ϕ)𝑑x,
Ωf(U)𝑑xγnBrΩf(ϕ)𝑑x,

where

γ=(|Ω||BrΩ|)1n.

For a proof, see [17] or [3] and, in particular, see [7] for a proof of (3.4).

The following observation will be useful in the sequel.

Remark 3.2.

Since U is constant on Ω (U=U(Ω)) and since ϕ is radial, we deduce that

ΩU2𝑑S=U2(Ω)|Ω|=ϕ2(rΩ)|BrΩ||Ω||BrΩ|=|Ω||BrΩ|BrΩϕ2𝑑S.

Since, by (3.1) we have |BrΩ||Ω|, the isoperimetric inequality implies that

|Ω||BrΩ|1.

Consider first the case λ1σ(Ω) with

σ<σ0(Ω)=-|Ω||Ω|.

Define

σ:=σ|Ω||BrΩ|σ.

Let u be a positive normalized radial eigenfunction of (1.1) in BrΩ with σ replaced by σ, corresponding to the eigenvalue λ1σ(BrΩ). Since by (3.1) we have

σ=-|σ||Ω||BrΩ|<σ0(Ω)|Ω||BrΩ|=-|Ω||BrΩ|σ0(BrΩ),

then u is of constant sign. Then, the transplanted function U of u in Ω satisfies

Ω|U|2𝑑x=1.

By (2.2) we have

1λ1σ(Ω)ΩU2𝑑x+σΩU2𝑑S.

Taking into account (3.3) and Remark 3.2, we get

1λ1σ(Ω)BrΩu2𝑑x-|σ||Ω||BrΩ|BrΩu2𝑑S.

By (3.6) the right-hand side is positive and equal to

1λ1σ(BrΩ).

Consequently,

0λ1σ(Ω)λ1σ(BrΩ).

Consider now the case σ0<σ<0. Define

σ′′:=σ|Ω||BrΩ||Ω||BrΩ|>-|BrΩ||BrΩ|=σ0(BrΩ).

Let u be a positive normalized radial eigenfunction of (1.1) in BrΩ, corresponding to the eigenvalue λ-1σ′′(BrΩ). Let U be the transplanted function of u in Ω. Then, by (2.3) we get

1λ-1σ(Ω)ΩU2𝑑x-|σ|ΩU2𝑑S.

We apply (3.4) to the first integral on the right-hand side and again Remark 3.2 to the second one. Thus,

1λ-1σ(Ω)γnBrΩu2𝑑x-|σ||Ω||BrΩ|Ωu2𝑑S=γn(BrΩu2𝑑x+σ′′Ωu2𝑑S).

Thus,

0>λ-1σ(Ω)1γnλ-1σ′′(BrΩ).

We may rewrite this inequality as

|Ω|λ-1σ(Ω)|BrΩ|λ-1σ′′(BrΩ).

The results of this section are summarized in the following theorem.

Theorem 3.3.

Let λ±1σ(Ω) be the first positive (negative) eigenvalue of (1.1) given by (2.1). Let rΩ be the harmonic radius of Ω and let σ and σ′′ be defined in (3.5) and (3.7), respectively. Then, the following inequalities hold.

  1. (i)If σ<σ0(Ω)<0, then 0λ1σ(Ω)λ1σ(BrΩ).
  2. (ii)If σ0(Ω)<σ<0, then 0>|Ω|λ-1σ(Ω)|BrΩ|λ-1σ′′(BrΩ).

Equality holds in both cases if and only if Ω is a ball.

Remark 3.4.

It is interesting to compare Theorem 3.3 (i) with (2.5) in Lemma 2.4 (i). We get the following two-sided bound: if BR is a ball of equal volume as Ω and if σ<σ0(BR)<0, then

λ1σ(BR)λ1σ(Ω)λ1σ(BrΩ),

where equality holds for the ball.

4 Domain dependence

4.1 Small perturbations of a given domain

We are interested in deriving optimality conditions for the domain functionals λ±1(Ω). Contrary to the results in [2] (see (2.5) and (2.6)), these results will be local. We first describe the general setting.

Consider a family of domains (Ωt)t. The parameter t varies in some open interval (-t0,t0), where t0>0 is prescribed. With this notation, we set Ω0:=Ω. The family is given by the following construction. Let

Φt:ΩΩt:=Φt(Ω)

with

Φt(x)=x+tv(x)+t22w(x)+o(t2)

be a smooth family, where v and w are vector fields such that

v,w:Ω¯nare in C1(Ω¯).

Note that, for t0>0 small enough, (Φt)|t|<t0 is a family of diffeomorphisms. This restricts t0 and defines the notion of “small perturbations of Ω”.

The volume of Ωt is given by

V(t):=|Ωt|=ΩJ(t)𝑑x,

where J(t) is the Jacobian determinant corresponding to the transformation Φt. The Jacobian matrix corresponding to this transformation is, up to second-order terms, of the form

I+tDv+t22Dw,

where

(Dv)ij=jviandj=xj.

By Jacobi’s formula, for small t, we have

J(t):=det(I+tDv+t22Dw)=1+tdivv+t22((divv)2-Dv:Dv+divw)+o(t2).

Here, we used the notation

Dv:Dv:=ivjjvi,

where the Einstein summation convention is used. Hence,

V(t)=BRJ(t)𝑑x=|Ω|+tΩdivvdx+t22Ω((divv)2-Dv:Dv+divw)𝑑x+o(t2).

For the first variation, we only have to require that Φt is volume preserving of the first order, that is,

V˙(0)=Ωdivvdx=Ω(vν)𝑑S=0.

We also consider perturbations which, in addition to (4.1), satisfy the volume preservation of the second order, namely,

V¨(0)=Ω((divv)2-Dv:Dv+divw)dx=0.

In addition, we consider perturbations of the surface area S(t)=|Ωt|. It can be expressed as an integral over Ω in the form

S(t)=Ωm(t)𝑑S.

An explicit form for m(t) is given in [9, (2.17)]. An expansion in t gives

S(t)=Ωm(t)𝑑S=|Ω|+tΩm˙(0)𝑑S+t22Ωm¨(0)𝑑S+o(t2).

Consider now perturbations Ωt which have the same area as Ω. As before, we say that ϕt is area preserving of the first order (cf. [8]) if

S˙(0)=Ω(vν)HΩ𝑑S=0,

where HΩ denotes the mean curvature of Ω. The perturbations satisfy the area-preserving condition of the second order if

S¨(0)=ΩF(*v,v)𝑑S+(n-1)Ω(wν)HΩ𝑑S=0.

This formula was derived in [8]. Here, F(*v,v) is a known scalar function of the tangential derivative *v of v (see, for example, [8, (2.20)]). If Ω=BR, this condition becomes

S¨(0):=BR|*(vν)|2𝑑S-n-1R2BR(vν)2𝑑S+n-1RV¨(0)=0,

where the last term in the sum is computed in (4.2) (see also [8, Lemma 2]). For later use, we set

S¨0(0):=BR|*(vν)|2𝑑S-n-1R2BR(vν)2𝑑S.

Thus,

S¨(0)=S¨0(0)+n-1RV¨(0).

Note that, for non-trivial volume-preserving perturbations, S¨(0)(=S¨0(0)) is strictly positive (see also [8, Section 7]).

4.2 First and second domain variation

4.2.1 First variation and monotonicity

Let (Ωt)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 (4.1) and (4.2) or area preserving in the sense of (4.3) and (4.4). For the moment, we denote by λ either of the two first eigenvalues λ±1. We denote by ut:Ωt the solution of

{Δut+λ(Ωt)ut=0in Ωt,νtut=λ(Ωt)σuton Ωt.

Here, λ(Ωt) has the representation

λ(t):=λ(Ωt)=Ωt|ut|2𝑑yΩtut2𝑑y+σΩtut2𝑑St.

Consequently, the energy is

(t)=Ωt|ut|2𝑑y-λ(t)(Ωtut2𝑑y+σΩtut2𝑑St)0for all t.

In [8], and in more detail in [9], the first and second variation of with respect to t were computed. For the first variation, we obtained (see [8, (4.1)])

0=˙(0)=Ω(vν)(|u|2-λu2-2λ2σ2u2-λσ(n-1)HΩu2)𝑑S-λ˙(0)(Ωu2𝑑x+σΩu2𝑑S),

where u(x) is the solution of (4.6) in Ω, that is, u=u0.

Remark 4.1.

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, for example, [19, Chapter IV, Section 3.5]). It was shown in Remark 2.1 that this is the case for λ1(t) if σ<σ0(Ωt) and for λ-1(t) if σ0(Ωt)<σ<0. By our regularity assumptions on the perturbations we have σ0(Ωt)σ0(Ω) as t0. Hence, for small t, we have that λ1 and λ-1 are differentiable.

The condition λ˙(0)=0 and (4.8) give the necessary condition

Ω(vν)(|u|2-λu2-2λ2σ2u2-λσ(n-1)HΩu2)𝑑S=0.

In the case of volume-preserving perturbations, (4.1) implies

|u|2-λu2-2λ2σ2u2-λσ(n-1)HΩu2=constanton Ω.

This is a special case of [8, Theorem 1]. In the case of area-preserving perturbations, we apply (4.3) and obtain

|u|2-λu2-2λ2σ2u2-λσ(n-1)HΩu2=constant×HΩon Ω

as a necessary condition for any critical point of λ(Ω). It is an open question whether (4.9) or (4.10) imply that Ω can only be a ball.

From now on, let Ω=BR. To ensure the differentiability of λ in t=0, we only consider the cases

λ=λ1if σ<σ0(BR)
andλ=λ-1if σ0(BR)<σ<0.

Then, (4.9) and (4.10) are satisfied since the corresponding eigenfunctions are radial. Hence, λ˙(0)=0, that is, the ball is a critical domain with respect to volume-preserving and area-preserving perturbations.

Formula (4.8) implies the monotonicity of λ±1 for nearly spherical domains with respect to volume-increasing (volume-decreasing) perturbations. Indeed, we rewrite (4.8) as

a(u,u)λ˙(0)=BR(ur2(R)-λu2(R)-2λ2σ2u2(R)-λσn-1Ru2(R))(vν)𝑑S.

Then, we use the boundary condition ur2(R)=λ2σ2u2(R) and we obtain

a(u,u)λ˙(0)=-u(R)k(R)BR(vν)𝑑S,

where

k(R):=λu(R)(1+(n-1)σR+λσ2).

Next, we determine the sign of k(R). To this end, we modify the proof of [7, Lemma 3]. For the sake of completeness, we give the details.

Lemma 4.2.

Let k(R) be given by (4.14) and let u(r) be the positive radial function in the case λ=λ1 or λ=λ-1. Then, we have

k(R)>0if λ=λ1 and σ<-Rn
𝑎𝑛𝑑k(R)<0if λ=λ-1 and -Rn<σ<0.

Proof.

In the radial case, either eigenfunction satisfies the differential equation

{urr+n-1rur+λu(r)=0in (0,R),u(R)=λσu(R).

We set

z=uru

and we observe that

dzdr+z2+n-1rz+λ=0in (0,R).

At the endpoint we have

dzdr(R)+λ2σ2+(n-1)Rλσ+λ=0.

We know that z(0)=0 and z(R)=λσ. Note that

zr(0)=-λ.

We distinguish two cases.

The case λ=λ-1(BR). In this case, we have (see also (4.15))

z(0)=0,z(R)=λ-1σ>0,zr(0)=-λ-1>0.

Thus, z(r) increases near 0. We again determine the sign of zr(R). If zr(R)0, then, because of (4.16), there exists a number ρ(0,R) such that zr(ρ)=0, z(ρ)>0 and zrr(ρ)0. From the equation we get

zrr(ρ)=n-1ρ2z(ρ)>0,

which is contradictory. Consequently,

zr(R)=-(λ-12σ2+(n-1)Rλ-1σ+λ-1)>0.

This also implies k(R)<0 in the case λ=λ-1(BR).

The case λ=λ1(BR). We have (also from (4.15))

z(0)=0,z(R)=λ1σ<0,zr(0)=-λ1<0.

Thus, z(r) decreases near 0. We determine the sign of zr(R). If zr(R)0, then, because of (4.17), there exists a number ρ(0,R) such that zr(ρ)=0, z(ρ)<0 and zrr(ρ)0. From the equation we get

zrr(ρ)=n-1ρ2z(ρ)<0,

which leads to a contradiction. Consequently,

zr(R)=-(λ12σ2+(n-1)Rλ1σ+λ1)<0.

This implies k(R)>0 in the case λ=λ1(BR). ∎

We easily prove the following lemma.

Lemma 4.3.

The first derivative λ˙±1(0) satisfies the sign condition

λ˙±1(0)<0if BR(vν)𝑑S>0
𝑎𝑛𝑑λ˙±1(0)>0if BR(vν)𝑑S<0.

Proof.

This follows directly from (4.13) and Lemma 4.2. Indeed, it is sufficient to recall that a(u,u)>0 for λ=λ1(BR) and a(u,u)<0 for λ=λ-1(BR). Also, note that u is positive in both cases. ∎

4.2.2 Second variation

We are interested in the extremal properties of the ball. Let ut be the solution of (4.6). Its pullback to Ω is given by u~(x,t)=ut(ϕt(x);t). We set u(x)=tut(;t)|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. We thereby use the fact that BR is a critical domain, that is, λ˙(0)=0. This follows the technique in [7]. Another good reference, with a slightly different approach, is the book of Henrot and Pierre [16]. For our discussion, we need the solution of the boundary value problem

{Δu+λu=0in BR,νu-λσu=k(R)(vν)on BR,

where k(R) is given in (4.14). To (4.18) we associate the quadratic form

Q(u):=BR|u|2𝑑x-λBR(u)2𝑑x-λσBR(u)2𝑑S.

We now turn to the computation of λ¨(0). We repeat the computations carried out in [8, Section 7]. We consider perturbations which are either volume or area preserving. In both cases, we restrict ourselves to the case Ω=BR, which implies that λ˙(0)=0. In a first step, we write (4.7) as

(t)=1(t)-λ(t)2(t).

Since (t)=0 by (4.7) and since λ˙(0)=0, differentiation with respect to t leads to

λ¨(0)=¨1(0)-λ(0)¨2(0)2(0).

Then, keeping in mind that 2(0)=a(u,u), we get the modified formula

λ¨(0)a(u,u)=-2Q(u)-λσu2(R)S¨0(0)-k(R)u(R)V¨(0)-2λσk(R)u(R)BR(vν)2𝑑S.

We note that the ball BR is a local minimizer for λ if

  1. (i)λ=λ1>0, σ<σ0(R)<0 and
    -2Q(u)-λ1σu2(R)S¨0(0)-k(R)u(R)V¨(0)-2λ1σk(R)u(R)BR(vν)2𝑑S>0;
  2. (ii)λ=λ-1<0, σ0(R)<σ<0 and
    -2Q(u)-λ-1σu2(R)S¨0(0)-k(R)u(R)V¨(0)-2λ-1σk(R)u(R)BR(vν)2𝑑S<0.

Finally, observe that for volume-preserving perturbations, we have V¨(0)=0, whereas for area-preserving transformations, we have S¨(0)=0. In the next section, we will discuss the sign of λ¨(0).

4.2.3 Sign of the second variation

We consider the Steklov eigenvalue problem

{Δϕ+λϕ=0in BR,νϕ-λσϕ=μϕon BR.

There exists an infinite number of eigenvalues

μ1<μ2μ3

with

limiμi=.

Remark 4.4.

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<σ0(BR) (respectively σ0(BR)<σ<0), the eigenvalue λ1 (respectively λ-1) is simple and the eigenfunction u1 (respectively u-1) is of constant sign. Thus, 0=μ=μ1. As a consequence, the spectrum only consists of non-negative eigenvalues.

There exists a complete system of eigenfunctions {ϕi}i1 such that

BRϕiϕj𝑑S=δij.

Similarly to the dicussion in [8, Section 7], we get the representation

u=i=2ciϕiand(vν)=i=2biϕi

for the solution u of (4.18) and the perturbation vν. Note that by (4.1) and (4.3) we have c1=b1=0, both in the case of volume-preserving and area-preserving perturbations.

It is easy to check that

Q(u)=i=2ci2μi.

The boundary condition in (4.18) implies

bi=ciμik(R),

thus

BR(vν)2𝑑SR=i=2ci2μi2k2(R),

where k(R) is defined in (4.14).

Volume-preserving perturbations.

In this case, we have V¨(0)=0 and by the isoperimetric inequality we have

S¨(0)=S¨0(0)+n-1RV¨(0)0.

If we exclude rotations and translations, we have the estimate

S¨0(0)n+1R2BR(vν)2𝑑S=n+1k2(R)R2i=2ci2μi2.

This was shown in the derivation of [8, (99)].

We first consider λ1. According to (4.19), a necessary condition for the ball to be a minimizer is

-2Q(u)-λ1σu2(R)S¨0(0)-2λ1σk(R)u(R)BR(vν)2𝑑S>0.

If we express the first and last terms by (4.22) and (4.23), we obtain the condition

2i=2ci2μi2(-1μi+λ1|σ|u(R)k(R))-λ1σu2(R)S¨0(0)>0.

Since λ1>0 and σ<0 and in view of (4.24), it is sufficient to show that the quantity in parentheses in the sum is positive. This follows from the next lemma.

Lemma 4.5.

We consider the eigenvalue λ1 given in (2.1). Let k(R) be given by (4.14) and let μ2 be the smallest positive eigenvalue μ2 of the Steklov eigenvalue problem (4.21). Then, we have

-1μ2+λ1|σ|u(R)k(R)>0.

Proof.

The idea is to compute μ2. To this end, we consider (4.21) for μ=μ1=0 and λ>0. Separation of variables, that is, ϕ(x)=w(r)φ(θ) with 0<r<R and θB1, yields the following results.

  1. (i)The function φ is an eigenfunction of the Laplace–Beltrami operator on B1. That is,
    Δ*φ+k(k+n-2)φ=0on B1
    for k=0,1,2,.
  2. (ii)The radial function w is a solution of
    w′′(r)+n-1rw(r)+(λ-k(k+n-2)r2)w(r)=0for 0<r<R
    with w(0)=0 and w(R)=λσw(R). Since λ>0, the solution is given by the regular Bessel functions of first type, that is,
    w(r)=rωJk+ω(λr),
    where ω=n-22. The eigenvalues λ are given by the boundary condition. Hence,
    λJk+ω(λR)Jk+ω(λR)=λσ+ωR.
    If we set k=0, we obtain λ=λ1.

For μ=μ2>0, we note that the Steklov eigenfunction associated to μ2 changes sign. Hence, k>0. On the other hand, it is well known that

JνJν

is an increasing function in ν. Thus, μ2 is determined from

λJ1+ω(λR)J1+ω(λR)=λσ+μ2+ωR.

We recall the recurrence relations

J1+ω(λR)=Jω(λR)-ω+1λRJ1+ω(λR)
andJ1+ω(λR)=-Jω(λR)+ωλRJω(λR).

We replace J1+ω in (4.28) by the right-hand side of (4.29). This gives

λ(Jω(λR)J1+ω(λR)-ω+1λR)=λσ+μ2+ωR.

Then, we use (4.31) to replace

JωJ1+ω

in (4.32). We obtain

λ(-λJω(λR)Jω(λR)-ω+1R)-1=λσ+μ2+ωR.

We rearrange terms and we obtain

μ2=1-σ(1+n-1Rσ+λ1σ2)=1|σ|k(R)λ1u(R)>0,

where we have used (4.14), Lemma 4.2, the positivity of the radial eigenfunction for λ1 and the fact that σ is negative. Finally, we insert this expression for μ2 into (4.26) and this gives the claim. ∎

The last lemma proves (4.25).

Theorem 4.6.

For some given ball BR let σ0(BR)=-Rn and let λ1 be given by (2.1). Then, the following optimality result holds: if σ<σ0(BR), then among all smooth domains of equal volume, the ball BR is a local minimizer for λ1. If we exclude translations and rotations of BR, then it is a strict local minimizer.

The case λ=λ-1(BR)<0 and σ0(R)<σ<0 is more involved. In that case, we have k(R)<0 by Lemma 4.2. Since V¨(0)=0, we deduce from (4.20) the necessary condition

2Q(u)+λ-1σu2(R)S¨0(0)+2λ-1σk(R)u(R)BR(vν)2𝑑S>0

for the ball to be a minimizer. In this case, note that λ-1σ>0. It is now crucial to apply (4.24). As a consequence, (4.33) holds if

2Q(u)+(λ-1σu2(R)n+1R2+2λ-1σk(R)u(R))BR(vν)2𝑑S>0.

In view of (4.22) and (4.23), this is equivalent to

2i=2ci2μi2(1μi+(n+1)λ-1σu2(R)2R2k2(R)+λ-1σu(R)k(R))>0.

Note that the first term in the sum is positive, as is the second one, while the third term is negative. Consequently, it suffices to show that

(n+1)λ-1σu2(R)2R2k2(R)+λ-1σu(R)k(R)>0.

This is equivalent to

n+1R2+2λ-1(1+n-1Rσ+λ-1σ2)>0.

A more detailed analysis shows that this quadratic expression in λ-1 has two negative zeros. Thus, we need bounds on λ-1 to verify (4.34). To this end, we proceed as in the case k=0 in the proof in Lemma 4.5. Instead of (4.27), for k=0, we get

λ-1RIω(λ-1R)Iω(λ-1R)=λ-1σR+ω,

where again ω=n-22 and the function I denotes the modified Bessel function of first type. In [10, Theorem B], the following estimate for the logarithmic derivative of I was proved: for all x>0 and ν>0, there holds

νν+1x2+ν2<xIν(x)Iν(x)<x2+ν2.

Thus, we obtain a two-sided bound for the right-hand side of (4.35), that is,

(1-2n)R2-(n-2)|σ|Rσ2R2<|λ-1|<R2-(n-2)|σ|Rσ2R2.

Consequently, there exists a number t[0,1] such that

|λ-1|:=(1-t2n)R2-(n-2)|σ|Rσ2R2.

Moreover, for 0<s<1, we set

|σ|=:sRn.

Then, an explicit computation shows that (4.34) is equivalent to

f(s,t):=-4t(n-2t)t+(2n-12t+4nt)s-(n-5)s2>0.

We want this inequality to hold for all 0<s<1 and 0t1. The function f has two zeros Z1(t) and Z2(t) parametrized over 0t1 with

Z1(t):=-6t+n+2nt+-4t2+8nt-16nt2+n2+4n2t2n-5
andZ2(t):=-6t-n-2nt+-4t2+8nt-16nt2+n2+4n2t2n-5.

Note that the root is real.

We will discuss the sign of f for n=2,3,.

  1. (i)The case n=5 is special. In this case, it turns out that, for all 0t1, we have f>0 if s0<s<1 with
    s0=52(2-3)0.67.
  2. (ii)In the case n=2,3,4, we compute that Z1(t)<0<Z2(t)<1 for 0t1. Thus, (4.36) holds if
    s>s0:=max{Z2(t):0t1}.
  3. (iii)In the case n6, the roles of Z1 and Z2 are interchanged. Now, we get 0<Z2(t)<1<Z1(t) for 0t1. Note also that in (4.36), the coefficient of s2 changes sign. Thus, (4.36) holds if again
    s>s0:=max{Z2(t):0t1}.

A direct computation yields an increasing sequence s0=s0(n) with

limns0(n)=-5+30.76.

We summarize our results in the following theorem.

Theorem 4.7.

For some given ball BR, let σ0(BR)=-Rn, let λ-1 be given by (2.1) and let s0:=-5+3. Then, the following optimality result holds: if σ0(BR)<σ<s0σ0(BR)<0, then among all smooth domains of equal volume, the ball BR is also a strict local minimizer for λ-1.

Area-preserving perturbations.

In this case, S¨(0)=0 and, therefore, by (4.5) we have

V¨(0)=-Rn-1S¨0(0)<0.

We first consider λ1. According to (4.19) and (4.37), a sufficient condition for the ball to be a minimizer is

-2Q(u)-λ1σu2(R)S¨0(0)+Rn-1k(R)u(R)S¨0(0)-2λ1σk(R)u(R)BR(vν)2𝑑SR>0.

If we compare this with (4.25), we see that an extra positive term occurs. Since the arguments in [8, Section 7.2.2] carry over to the area-preserving case without any changes, we have proved the following theorem.

Theorem 4.8.

For some given ball BR, let σ0(BR)=-Rn and let λ1 be given by (2.1). Then, the following optimality result holds: if σ<σ0(BR), then among all smooth domains of equal area, the ball BR is a strict local minimizer for λ1.

The case λ=λ-1(BR)<0 and σ0(R)<σ<0 is similar. In that case, k(R)<0 by Lemma 4.2. Since (4.37) holds, we deduce from (4.20) the sufficient condition

2Q(u)+λ-1σu2(R)S¨0(0)+k(R)u(R)V¨(0)+2λ-1σk(R)u(R)BR(vν)2𝑑S>0

for the ball to be a minimizer. In this case, note that λ-1σ>0, thus we can apply (4.24) again. As a consequence, (4.38) holds if

2Q(u)-Rn-1k(R)u(R)S¨0(0)+(λ-1σu2(R)n+1R2+2λ-1σk(R)u(R))BR(vν)2𝑑S>0.

Again, an additional positive term occurs. Inequality (4.34) now reads

n+1R2+(2λ-1-n+1n-11σR)(1+n-1Rσ+λ-1σ2)>0.

Using the same arguments, we obtain a modified version of (4.36), that is,

f(s,t):=-(4(n-1))(n-2t)t+(2n2-2n-14tn+14t+4tn2)s-(n-2)(n-3)s2>0.

We compute the zeros of f and we obtain analogous expressions for Z1 and Z2. Repeating the computations done in (i)–(iii), we get a sequence (s~(n))n of lower bounds for s such that f>0. Then, the extra positive term in the second variation implies s~(n)<s(n) for all n, however the asymptotic behavior is the same, that is,

limns0(n)=-5+30.76.

Theorem 4.9.

For some given ball BR, let σ0(BR)=-Rn, let λ-1 be given by (2.1) and let s0:=-5+3. Then, the following optimality result holds: if σ0(BR)<σ<s0σ0(BR)<0, then among all smooth domains of equal volume, the ball BR is a strict local minimizer for λ-1.

5 Open problems

Problem 1.

The variational characterization of λ±1 (see (2.1)) is also related to two inequalities, known as Friedrich’s inequality and the trace inequality. In fact, for σ<σ0<0, we get Friedrich’s inequality

Ωu2𝑑x1λ1(Ω)Ω|u|2𝑑x+|σ|Ωu2𝑑S

and, for σ0<σ<0, we get the trace inequality

Ωu2𝑑S1σλ-1(Ω)Ω|u|2𝑑x+1|σ|Ωu2𝑑x.

The second inequality was also considered in [1], where the case of equality was analyzed.

It is an interesting open problem to find explicit lower bounds for λ1(Ω) and σλ-1(Ω). Note that the technique of harmonic transplantation gives upper bounds for these two quantities.

Problem 2.

At least for λ-1(Ω), it may be true that balls are the only local minimizers among all nearly spherical smooth domains of given volume. There is no global result available at the moment. Therefore, motivated by Theorem 3.3 (ii), it may also be interesting to ask if a quantity like |Ω|λ-1(Ω) has the ball of equal volume as a minimizer.

Acknowledgements

This paper was written during a visit at the Newton Institute in Cambridge. Both authors would like to thank the Newton Institute for the excellent working atmosphere. The authors are also greatly indebted to the referee for his many suggestions which improved the presentation and for pointing out an error in the proof of Theorem 4.7.

References

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    C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), no. 1, 35–67.

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    C. Bandle, J. von Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), 73–104.

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    J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann. 284 (1989), 285–305.

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    G. François, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal. 46 (2006), no. 1, 43–52.

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    A. Henrot and M. Pierre, Variation et optimisation de formes, Springer, Berlin, 2005.

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    J. Hersch, Transplantation harmonique, transplantation par modules, et théorèmes isopérimétriques, Comment. Math. Helv. 44 (1969), 354–366.

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    T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989), no. 1–2, 43–60.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    G. Auchmuty, Sharp boundary trace inequalities, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), 1–12.

  • [2]

    C. Bandle, A Rayleigh–Faber–Krahn inequality and some monotonicity properties for eigenvalue problems with mixed boundary conditions, Inequalities and Applications, Internat. Ser. Numer. Math. 157, Birkhäuser, Basel (2009), 3–12.

  • [3]

    C. Bandle and M. Flucher, Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations ΔU = = e U e^{U} and ΔU = = U ( n + 2 ) / ( n - 2 ) U^{(n+2)/(n-2)}, SIAM Rev. 38 (1996), 191–238.

  • [4]

    C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), no. 1, 35–67.

  • [5]

    C. Bandle, J. von Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), 73–104.

  • [6]

    C. Bandle and W. Reichel, A linear parabolic problem with non-dissipative dynamical boundary conditions, Recent Advances on Elliptic and Parabolic Issues, World Scientific, Hackensack (2006), 45–77.

  • [7]

    C. Bandle and A. Wagner, Isoperimetric inequalities for the principal eigenvalue of a membrane and the energy of problems with Robin boundary conditions, J. Convex Anal. 22 (2015), no. 3, 627–640.

  • [8]

    C. Bandle and A. Wagner, Second domain variation for problems with Robin boundary conditions, J. Optim. Theory Appl. 167 (2015), no. 2, 430–463.

  • [9]

    C. Bandle and A. Wagner, Second variation of domain functionals and applications to problems with Robin boundary conditions, preprint (2015), http://arxiv.org/abs/1403.2220.

  • [10]

    A. Baricz, S. Ponnusamy and M. Vuorinen, Functional inequalities for modified Bessel functions, Expo. Math. 29 (2011), no. 4, 399–414.

  • [11]

    J. Ercolano and M. Schechter, Spectral theory for operators generated by elliptic boundary problems with eigenvalue parameter in boundary conditions. I, Comm. Pure Appl. Math. 18 (1965), 83–105.

  • [12]

    J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann. 284 (1989), 285–305.

  • [13]

    J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions, Progress in Partial Differential Equations: The Metz Surveys 2, Pitman Res. Notes Math. Ser. 296, Longman Scientific & Technical, Harlow (1993), 138–148.

  • [14]

    J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations 18 (1993), 1309–1364.

  • [15]

    G. François, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal. 46 (2006), no. 1, 43–52.

  • [16]

    A. Henrot and M. Pierre, Variation et optimisation de formes, Springer, Berlin, 2005.

  • [17]

    J. Hersch, Transplantation harmonique, transplantation par modules, et théorèmes isopérimétriques, Comment. Math. Helv. 44 (1969), 354–366.

  • [18]

    T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989), no. 1–2, 43–60.

  • [19]

    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.

  • [20]

    J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm. Partial Differential Equations 33 (2008), no. 4–6, 561–612.

  • [21]

    J. L. Vázquez and E. Vitillaro, On the Laplace equation with dynamical boundary conditions of reactive-diffusive type, J. Math. Anal. Appl. 354 (2009), no. 2, 674–688.

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