Asymptotic behaviour of the Steklov problem on dumbbell domains

HAL Id: hal-02895481

https://hal.archives-ouvertes.fr/hal-02895481

Submitted on 9 Jul 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Asymptotic behaviour of the Steklov problem on dumbbell domains

Dorin Bucur, Antoine Henrot, Marco Michetti

To cite this version:

Dorin Bucur, Antoine Henrot, Marco Michetti. Asymptotic behaviour of the Steklov problem on dumbbell domains. Communications in Partial Differential Equations, Taylor & Francis, 2021, 46 (2), pp.362-393. �10.1080/03605302.2020.1840587�. �hal-02895481�

DUMBBELL DOMAINS

DORIN BUCUR, ANTOINE HENROT, MARCO MICHETTI

Abstract. We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.

1. Introduction

The purpose of this paper is to analyse the asymptotic behaviour of the eigenvalues and eigenfunctions of the Steklov problem in a dumbbell domain. Given Ω⊆ Rⁿ, open, bounded, connected, Lipschitz set, the Steklov problem on Ω consists in solving the eigen- value problem

(1)

(∆u= 0 Ω

∂_νu=σu ∂Ω,

whereν stands for the outward normal at the boundary. As the trace operator H¹(Ω)→ L²(∂Ω) is compact, the spectrum of the Steklov problem is discrete and the eigenvalues (counted with their multiplicities) go to infinity

0 =σ0(Ω)< σ1(Ω)≤σ2(Ω)≤ · · · →+∞.

We also have the following variational characterization of the Steklov eignevalues σ_k(Ω) = inf

E_k sup

06=u∈E_k

R

Ω|∇u|²dx R

∂Ωu²dHⁿ⁻¹,

where the infimum is taken over allk-dimensional subspaces of the Sobolev space H¹(Ω) which areL²-orthogonal to constants on∂Ω.

Let Ω⊂Rⁿ be a dumbbell shape domain given by (see Figure 1) Ω=D1∪T∪D2,

whereD1 andD2 are disjoint, bounded, open, connected sets inRⁿwith Lipschitz bound- ary andT is expressed as

T=

x= (x1, x⁰)∈Rⁿ| − L

2 ≤x1 ≤ L

2,|x⁰|< ρ(x1) , whereL >0 andρ∈C⁰([−^L₂,^L₂])∩C^∞((−^L₂,^L₂)) is a positive function.

1

D₁ T D₂

Figure 1. Dumbbell shape domain Ω.

The connection between the channel and the two regions D1 and D2 occurs as follows:

we assume that there exist an orthogonal system of coordinate x = (x₁, x₂, ..., x_n) = (x1, x⁰)∈Rⁿ and two constantsL, δ∈Rsuch that

D₁∩

x= (x₁, x⁰)∈Rⁿ|x₁≥ −L

2,|x⁰| ≤δ =

x= (−L

2, x⁰)∈Rⁿ| |x⁰| ≤δ D₂∩

x= (x₁, x⁰)∈Rⁿ|x₁ ≤ L

2,|x⁰| ≤δ = x= (L

2, x⁰)∈Rⁿ| |x⁰| ≤δ . The eigenvalues of the Steklov problem in Ω are denoted by

0 =σ₀ < σ₁ ≤σ₂≤...% ∞ ∀ >0,

multiplicity being counted, and the corresponding eigenfunctions by u_k, which are nor- malized inL²(∂Ω),||u_k||_L2(∂Ω)= 1.

The main purpose of this work is to study what is the behaviour of (σ_k, u_k) whengoes to 0. The first thing to notice is that, if →0, the channel T collapse to a line and the norm of the trace operator blows up. One can easily observe that

∀k∈N, σ_k →0 when →0,

our objective being to give precise estimates of the asymptotic behaviour of σ_k when goes to 0. We shall prove thatσ_kbehaves, roughly speaking, asµk^γ, whereµkis thek-th eigenvalue of some one dimensional problem andγ ∈ {1, n−1}.

As an interesting feature, we notice that the behaviour strongly depends on the dimen- sion of the ambient space. Indeed we have to distinguish between the cases n = 2 and n≥3, as we shall see below. This fact is due to the presence of the boundary energy in the Rayleigh quotient of the Steklov problem and to the fact that in dimension three, or higher, the surface area measure of the boundary of the tube is vanishing with.

Below, we denote byP(D) the surface area measure of the boundary ofDandωnis the Lebesgue measure of then−dimensional unit ball. Let Φ_ε:T₁→T, Φ(x₁, x⁰) = (x₁, x⁰).

Here are our main results. The first theorem concerns the case n= 2.

Theorem 1.1. (n= 2)LetΩ ⊂R² be the dumbbell shape domain defined as above. Then σ_k ∼µ_k+o() as →0,

where µ_k is the k−th eigenvalue of the following problem

(2)











−_dx^d ρ(x)^dV_dx^k(x)

=µ_kV_k(x) x∈ −^L₂,^L₂ ρ(−^L₂)^dV_dx^k(−^L₂) =−^µ₂^kP(D₁)V_k(−^L₂)

ρ(^L₂)^dV_dx^k(^L₂) = ^µ₂^kP(D2)V_k(^L₂).

For every subsequence {_n}^∞_n=1 such that n→0, we have u_kⁿ◦Φn * Vk in H¹(T1),

where V_k is a k−th eigenfunction of the problem (2) constantly extended in the variable x2.

This kind of eigenvalue problem (in any dimension) where the eigenvalue µk appears both inside the domain and in the boundary condition is sometimes called a dynamical eigenvalue problem. It appears at different places in the literature. we refer for example to [23] where a complete study of this eigenvalue problem has been done. See also [9] where a similar problem appears in the homogenization of the Steklov problem.

The next two theorems concern the case n ≥ 3. We shall distinguish between the behaviour of the first non-zero eigenvalue, and the others.

Theorem 1.2. (n ≥ 3, k ≥ 2) Let Ω ⊂ Rⁿ be the dumbbell shape domain defined as above and n≥3. Then for all k≥2 we have

σ_k∼αk−1+o() as →0, where αk−1is the (k−1)−th eigenvalue (counting from zero) of

(3)











−wn−1 d

dx ρⁿ⁻¹(x)^dV_dx^k(x)

=αkwn−2ρⁿ⁻²(x)Vk(x) x∈ −^L₂,^L₂ V_k(−^L₂) = 0

V_k(^L₂) = 0.

For every subsequence {_n}^∞_n=1 such that _n→0, we have

n−2

n2 u_kⁿ◦Φn * Vk−1 in H¹(T1),

where Vk−1 is an eigenfunction corresponding to αk−1, constantly extended into the vari- ables xi for 2≤i≤n.

Therefore, in the case n ≥ 3, k ≥ 2 we end up with a classical Dirichlet eigenvalue problem.

Theorem 1.3. (n ≥ 3, k = 1) Let Ω ⊂ Rⁿ be the dumbbell shape domain defined as above and n≥3. The first Steklov eigenvalue has the following asymptotic behaviour

σ₁ ∼σ₁ⁿ⁻¹+o(ⁿ⁻¹) as →0,

where σ₁ is the unique positive number such that the following differential equation has a non-trivial solution:

(4)











−ω_n−1dx^d ρⁿ⁻¹(x)^dV_dx¹(x)

= 0 x∈ − ^L₂,^L₂ ρⁿ⁻¹(−^L₂)^dV_dx¹(−^L₂) =−_ω^σ1

n−1P(D1)V1(−^L₂) ρⁿ⁻¹(^L₂)^dV_dx¹(^L₂) = _ω^σ1

n−1P(D₂)V₁(^L₂).

For every subsequence {_n}^∞_n=1 such that _n→0, we have u₁ⁿ◦Φ_n * V₁ in H¹(T₁),

where V1 is the solution of the equation (4) constantly extended to the variables xi for 2≤i≤n.

Let us now comment on the existing literature. A similar problem for the eigenvalues of the Neumann Laplacian has been deeply studied, in particular in a series of papers by S. Jimbo. A first characterization of the eigenvalues in the Neumann case was given in [3].

In [16] there is a complete description of the behaviour of the Neumann eigenfunctions and in [19] there is a complete description of the Neumann eigenvalues when the channel collapse to a segment. Other references for the Neumann problem in dumbbell shape domains are [1, 14, 15, 17, 18]. These results turn out to be very useful in the study of the solutions of reaction diffusion systems in singular domains (see for instance [2, 7, 12, 20]).

Perturbations of the geometric domain for the Steklov problem have been considered in [11]. For an asymptotic behaviour of the Steklov problem on a singular perturbation somehow close to our analysis, we refer to the result of Nazarov [21] where he studies a two dimensional domain obtained by the junction of two rectangles (see also [22] for a perturbation by a small whole). At last, let us mention that in the case of Dirichlet boundary conditions, singular perturbations of this type are less interesting, since the spectrum is stable to this geometric perturbation. Indeed, it can be proved that the dumbbellγ-converges to the union of the two setsD₁∪D₂ which means that its Dirichlet eigenvalues converge to the union of the spectrum ofD1 and D2. We refer to the books [4] and [13] for more details.

2. The case n= 2. Proof of Theorem 1.1.

In this section we will prove Theorem 1.1. We define ∂T^e ⊂∂Ω in the following way

∂T^e=

x= (x1, x⁰)∈Rⁿ| −L

2 ≤x1≤ L

2, x⁰ =|ρ(x₁)|

In two dimensions, the set ∂T^e is not connected and we decompose it

∂T^e= Γ⁻ ∪Γ⁺ , where

Γ⁺ =

x= (x₁, x₂)∈Rⁿ| − L

2 ≤x₁≤ L

2, x₂ =ρ(x₁) Γ⁻ =

x= (x1, x2)∈Rⁿ| − L

2 ≤x1≤ L

2, x2 =−ρ(x₁) .

2.1. Upper bound for Steklov eigenvalues. First of all we prove that there exists a constantC >0 such that the following upper bound holds for small enough:

(5) σ_k ≤C.

Precisely, we prove the following.

Lemma 2.1. Let µ_k the k−th eigenvalue of (2) then we have

(6) σ_k ≤µ_k+o().

Proof. In order to obtain this upper bound, we use the variational formulation σ_k = inf

Ek

sup

06=u∈E_k

R

Ω|∇u|²dx R

∂Ωu²ds ,

where the infimum is taken over allk−dimensional subspace of the Sobolev spaceH¹(Ω) which are orthogonal to constants on ∂Ω. We choose a particular subspace E_k in order to obtain the upper bound.

We consider the eigenvalue problem (2) and take a basis of eigenfunctions {φ_i}_i∈N normalized in the following way

(7)

Z ^L

2

−L 2

φ_iφ_jdx₁+1

2P(D₂)φ_i L 2

φ_j L 2

+1

2P(D₁)φ_i −L 2

−L 2

=δ_ij. Then

Z ^L

2

−L 2

ρφ⁰_iφ⁰_jdx₁= 0 if i6=j (8)

Z ^L

2

−L 2

ρ(φ⁰_i)²dx1=µi. (9)

From the variational formulation of the eigenvalue problem we know that∀v∈H¹(−^L₂,^L₂) Z ^L

2

−L 2

ρφ⁰_iv⁰dx₁ = µi

2 P(D₂)φ_i L 2

v L 2

+µi

2P(D₁)φ_i −L 2

v − L 2

+µ_i Z ^L

2

−L 2

φ_ivdx₁,

we choosev= 1 we obtain:

(10) 1

2P(D₂)φ_i L 2

+1

2P(D₁)φ_i −L 2

+ Z ^L₂

−L 2

φ_idx₁= 0.

We now introduce our test functions that are the basis of our test subspace E_k. We define

Φi=











φ_i(−^L₂) if (x₁, x₂)∈D₁ φ_i(x₁) if (x₁, x₂)∈T φ_i(^L₂) if (x₁, x₂)∈D₂, and we introduce its mean value

m_i = 1

|∂Ω| Z

∂Ω

Φids.

The mean goes to zero if→0, indeed Z

∂Ω

Φ_ids=P(D₂)φ_i L 2

+P(D₁)φ_i −L 2

+ 2 Z ^L

2

−L 2

φ_ip

1 +²ρ⁰²dx₁,

from equation (10), dominated convergence and the fact that |∂Ω| →P(D1) +P(D2) + 2L >0 we obtain

(11) m_i →0 ∀i∈N.

We introduce now our basis elements

Ψi= Φi−m_i,

and our subspace will beE_k = Span<Ψ1, ...,Ψ_k >. Now we compute all the quantities we need for the Rayleigh quotient. We start by the numerator, ifi6=j:

Z

Ω

∇Ψ_i· ∇Ψ_jds= Z

T

∇Φ_i· ∇Φ_jds= 2 Z ^L

2

−L 2

ρφ⁰_iφ⁰_jdx1 = 0, where the last equality is given by (8), and

Z

Ω

|∇Ψ_i|²ds= Z

T

|∇Φ_i|²ds= 2 Z ^L₂

−L 2

ρ(φ⁰_i)²dx₁= 2µ_i,

where the last equality is given by (9). Now we compute the terms in the denominator, f_i,j() :=

Z

∂Ω

Ψ_iΨ_jds= Z

∂Ω

(Φ_i−m_i)(Φ_j−m_j)ds= Z

∂Ω

Φ_iΦ_jds−m_im_jP(Ω)

= 1

2P(D₂)φ_i L 2

φ_j L 2

+1

2P(D₁)φ_i − L 2

φ_j − L 2

+ Z ^L

2

−L 2

φ_iφ_jp

1 +²ρ⁰²dx₁−m_im_jP(Ω).

From (11), (7) and the dominated convergence we obtain

(12) lim

→0f_i,j() = 0 i6=j.

Similarly, f_i,i() :=

Z

∂Ω

Ψ²_ids= Z

∂Ω

(Φ_i−m_i)²ds= Z

∂Ω

Φ²_ids−(m_i)²P(Ω)

= 2 Z ^L

2

−L 2

φ²_ip

1 +²ρ⁰²dx₁+P(D₂)φ_i L 2

2

+P(D₁)φ_i −L 2

2

−(m_i)²P(Ω), now from (11), (7) and dominated convergence we obtain,

(13) lim

→0f_i,i() = 2.

Now if we use the test subspaceE_k in the variational characterization we obtain σ_k ≤ sup

(x1,...,xk)∈R^k

2Pk i=1x²_iµi

Pk

i=1x²_ifi,i() +P

i<j2xixjfi,j(),

ifis small enough from (12) and (13) we obtain

(14) σ_k ≤ sup

(x1,...,xk)∈R^k

Pk i=1x²_iµ_i Pk

i=1x²_i +o() =µ_k+o().

2.2. Convergence of eigenfunctions. We start by showing the convergence on the two regionsDi where i= 1,2

Lemma 2.2. Let k≥1 we have (up to a sub-sequence that we still denote by u_k) u_k* c_i,k in H¹(D_i),

u_k→c_i,k locally uniformly in D_i. where c_i,k ∈R are constants

Proof. First of all we know that σ_k → 0 as goes to 0, and from ||u_k||_L2(∂Ω) = 1 we conclude that

lim→0

Z

Ω

|∇u_k|²dx= 0,

so it means that ||∇u_k||_L2(D1) ≤ ||∇u_k||_L2(Ω) ≤ C. Now we want to bound ||u_k||_L2(D1)

uniformly on. Using Poincar´e-Friedrichs inequality we obtain Z

D1

(u_k)²dx≤ Z

Ω

(u_k)²dx≤C_Ω Z

Ω

|∇u_k|²dx+ Z

∂Ω

(u_k)²ds ,

we know that||u_k||_L2(∂Ω) = 1 and ||∇u_k||_L2(Ω) ≤C, we have only to check that C_Ω ≤ C≤ ∞ifis small enough.

We have the following variational characterization for the constant CΩ

1

C_Ω = inf

v∈H¹(Ω)

R

Ω|∇v|²dx+R

∂Ωv²ds R

Ωv²dx =λ₁(Ω,1),

whereλ₁(Ω,1) is the first Robin eigenvalue with boundary parameter 1 (see [6]). We de- note byBR the ball with the same measure of Ω, now, using the Bossel-Daners inequality and the rescaling property of the Robin eigenvalue (see [6]), we obtain

1

C_Ω =λ1(Ω,1)≥λ1(BR,1) = 1

R²λ1(B1, R).

Now, forsmall enough, we have|D₁|+|D₂| ≤ |Ω| ≤ |D₁|+|D₂|+ 1 so, by monotonicity of the Robin eigenvalue on balls we finally obtain

1

C_Ω =λ1(Ω,1)≥λ1(BR,1) = π

|D₁|+|D₂|+ 1λ1 B1,

r|D₁|+|D₂| π

>0.

Finally we conclude thatCΩ ≤C <∞for small enough. We conclude that

||u_k||_H1(D1)≤C <∞,

so exist a sequence, that we still denote byu_k, and u⁰_k∈H¹(D1) such that u_k* u⁰_k in H¹(D₁).

We also know that||∇u_k||_L2(D1)→0, so we conclude that there exists a constantc_1,k ∈R such that

u_k* c_1,k in H¹(D₁).

We can improve this convergence since u_k are harmonic. Fix a compact set K ⊂D1 and take δ > 0 such that B_δ(x) ⊂ D₁ for all x ∈ K. By the average properties of harmonic functions and the Cauchy-Schwartz inequality we have:

|u_k(x)|= 1

|B_δ(x)|

Z

B_δ(x)

|u_k(y)|dy≤ |B_δ(x)|⁻¹²||u_k||_L2(D1)≤C.

Up to a subsequence, we get thatu_kuniformly converges onK to a constant. We conclude that fori= 1,2

u_k * ci,k in H¹(Di),

u_k →ci,k locally uniformly in Di.

Now we study the behaviour of the eigenfunctions in the tubeT. We define the following functions

v_k(x₁, x₂) =u_k(x₁, x₂) ∀(x₁, x₂)∈T₁ Lemma 2.3. Let k≥1. There exists V_k∈H¹(T1) such that

v_k* Vk in H¹(T1),

(up to a sub-sequence, still denoted by v_k), where Vk depends only on the variablex1. Proof. We start with the bound of||∇v_k||_L2(T1)

Z

T1

|∇v_k|²dx≤ Z

T1

∂v_k

∂x₁ 2

+ 1 ²

∂v_k

∂x₂ 2

dx= 1

Z

T

|∇u_k|²dy≤C

where we did the change of coordinatesy₁ =x₁,y₂ =x₂ and the last inequality is true because of (5). We want now to bound||v_k||_L2(T1). By the Poincar´e-Friedrichs inequality we get

Z

T1

(v_k)²dx≤C_T1 Z

T1

|∇v_k|²dx+ Z

Γ⁺₁∪Γ⁻₁

(v_k)²ds .

Now||∇v_k||_L2(T1) is bounded, so it remains to bound the second term in the r.h.s. of the inequality. Sincep

1 +ρ⁰² is a bounded function, for small enough we obtain Z

Γ⁺₁

(v_k)²ds= Z ^L₂

−^L

2

(v_k(x₁, ρ(x₁))²p

1 +ρ⁰²dx₁

≤C Z ^L

2

−^L₂

(u_k(x1, ρ(x1))²p

1 +²ρ⁰²dx1

≤C Z

Γ⁺

(u_k)²ds≤C

where the last inequality is true becuase||u_k||_L2(∂Ω) = 1. The same computation is true for the integral over Γ⁻₁.

We conclude that there exists V_k ∈ H¹(T₁) such that (up to a sub-sequence that we still denote byv_k)

v_k * Vk in H¹(T1).

We finish the proof by showing thatV_k does not depend on x₂. Indeed Z

T1

∂v_k

∂x2

2

dx= Z

T

∂u_k

∂x2

2

dx≤C²→0.

2.3. Limit eigenvalue problem. From (6) we know that there exists 0≤β ≤µ_k such that

σ_k →β.

First, we prove that there exists j∈N such that 0≤j ≤k and β =µ_j and, in a second step, we prove thatj =k.

Step 1. We begin with the following.

Lemma 2.4. There exists j∈Nsuch that 0≤j≤k and σ_k

→µ_j, where µj is the j−th eigenvalue of the problem (2).

Proof. We define β in such a way that σ_k

→β,

from (6) we know that 0 ≤ β ≤ µk. We use the variational formulation of the Steklov problem with the following test function φ∈C_c^∞(−^L₂,^L₂) (we constantly extend φin the last variablex₂), we obtain:

Z

T

∂u_k

∂x1

∂φ

∂x1

dx=σ_k Z

Γ⁺∪Γ⁻

u_kφds.

Now we make the change of variable y1 = x1 and y2 =x2 and we write the integral in the right hand side by the integral in the graph ofρ

Z

T1

∂v_k

∂y1

∂φ

∂y1

dy= σ_k

Z ^L

2

−^L₂

u_k(x₁, ρ(x₁))φp

1 +²ρ⁰²dx₁+ Z ^L

2

−^L₂

u_k(x₁,−ρ(x₁))φp

1 +²ρ⁰²dx₁

= σ_k

Z ^L

2

−^L₂

v_k(x1, ρ(x1))φp

1 +²ρ⁰²dx1+ Z ^L

2

−^L₂

v_k(x1,−ρ(x₁))φp

1 +²ρ⁰²dx1

We know thatv_k * V_k inH¹(T₁) andV_k depends only on the variablex₁, we introduce the functionV_k that is the restriction of V_k to the variablex₁. We let goes to 0 and we obtain

Z

T1

∂Vk

∂y1

∂φ

∂y1

dy= 2β Z ^L

2

−^L₂

Vkφdx1.

Integrating by parts in the left hand side, we finally obtain

−2 Z ^L

2

−^L₂

d dx1

ρ d dx1

Vk

φdx1= 2β Z ^L

2

−^L₂

Vkφdx1.

This relation is true for every test function φ ∈ C_c^∞(−^L₂,^L₂) so we have that V_k and β must have to satisfy the following differential equation

(15) − d

dx ρ(x)dV_k dx(x)

=βV_k(x) x∈ −L 2,L

2 .

We find now the boundary conditions associated to this equation. Fix a real number ξ >0 and define the extended function ρ in the following way

ρ=











ρ(−^L₂) if −^L₂ −ξ ≤x1 ≤ −^L₂ ρ(x1) if − ^L₂ ≤x1 ≤ ^L₂ ρ(^L₂) if ^L₂ ≤x1≤ ^L₂ +ξ.

We define the extended tubeE

(16) E=

x= (x1, x2)∈R²| −L

2 −ξ ≤x1≤ L

2 +ξ,|x₂|< ρ(x1) ,

and we chooseξ in such a way thatE ⊂Ω. Now, repeating all the arguments in Lemma 2.3, we obtain that

v_k * V_k in H¹(E₁)

and V_k depends only on x₁. We also know from Lemma 2.2 that u_k locally uniformly converge toc1,k inD1. From this fact we have that

v_k(−L

2 −δ,0)→c1,k=Vk(−L

2 −δ) ∀ξ ≥δ >0,

where V_k is the restriction of V_k to the variable x₁. We know that V_k ∈ H¹(E₁), from embedding theoremVk is continuous so we finally obtain

(17) Vk(−L

2) =c1,k = lim

δ→0Vk(−L 2 −δ).

Similarly,

Vk(L

2) =c2,k.

We use the variational formulation of the Steklov eigenvalue with a test function ψ defined on all Ω and that depends only onx₁,

Z

Ω

∂u_k

∂x1

∂ψ

∂x1

dx=σ_k Z

∂Ω

u_kψds.

We repeat all the computations above and letting goes to 0, we obtain 2

Z ^L

2

−^L₂

ρdVk

dx1

dψ dx1

dx1=β

Vk −L 2

Z

∂D1

ψds+Vk

L 2

Z

∂D2

ψds+ 2 Z ^L

2

−^L₂

Vkψdx1

.

Integrating by parts the left hand side and recalling the equation (15) we finally get ρ(L

2)dV_k dx (L

2)ψ(L

2)−ρ(−L 2)dV_k

dx(−L

2)ψ(−L

2) =β V_k −L 2

Z

∂D1

ψds+V_k L 2

Z

∂D2

ψds .

Choosing a test function such thatψ= 1 inD1 andψ= 0 inD2, we get the first boundary condition

ρ(−L 2)dV_k

dx(−L

2) =−β

2P(D1)Vk(−L 2)

and, similarly, chosing a test function such that ψ= 0 in D₁ and ψ= 1 inD₂ we get the second boundary condition

ρ(L 2)dV_k

dx (L 2) = β

2P(D₂)V_k(L 2).

We finally obtain the following eigenvalue problem for β

(18)











−_dx^d ρ(x)^dV_dx^k(x)

=βV_k(x) x∈ − ^L₂,^L₂ ρ(−^L₂)^dV_dx^k(−^L₂) =−^β₂P(D₁)V_k(−^L₂)

ρ(^L₂)^dV_dx^k(^L₂) = ^β₂P(D2)V_k(^L₂).

To be able to conclude, it remains to prove thatVkis not the zero function. IfVkwould be zero, from the normalizationR

∂Ωu_k² = 1 and the convergence on the extended tube, we would have

1 =P(D₁)c²_1,k+P(D₂)c²_2,k+ 2 Z L/2

L/2

V_k².

Therefore, c_1,k or c_2,k would not be zero yielding a contradiction since the function V_k being inH¹(−^L₂−δ,^L₂+δ) is continuous and thus cannot be constant (different from zero) on (−^L₂−δ,−^L₂) and zero after−^L₂. Therefore we have proved that there existj∈Nsuch that 0≤j≤kand β=µ_j, where µ_j is the j−th eigenvalue of the problem (2).

Step 2. We have just proved that σ_k ∼µ_j, with 0≤j≤k. In this step we justify that j = k . We denote by Ve_k the function constructed by taking the k−th eigenfunction of problem (2) and extending it constantly in thex₂ variable and equally constant toV_k(−^L₂) inD₁ and to V_k(^L₂) in D₂.

We proceed by induction on the eigenvalue rank k. The case k = 0 is obvious. Now suppose that for all 0≤j ≤k−1 we have that v_j * Vj inH¹(T1) and σ_j∼µj.

Now we will prove that µk+o()≤σ_k. By contradiction we suppose that there exists j∈Nsuch that 0≤j≤k−1,v_k * Vj inH¹(T1) andσ_k ∼µj. From the orthogonality of the Steklov eigenfunctions we have the following equality

0 = lim

→0

Z

∂Ω

u_kVe_jds+ Z

∂Ω

u_k(u_j−Ve_j)ds.

For the first term, from (7), we have that lim→0R

∂Ωu_kVe_j = 2. For the second term, we recall the inductive hypothesisv_j* V_j, using the same argument in the proof of Lemma

2.4 we conclude also the equality (17) and, using Cauchy-Schwartz inequality, we obtain

|lim

→0

Z

∂Ω

u_k(u_j−Vj)ds| ≤lim

→0||u_k||_L2(∂Ω)||u_j−Vj||_L2(∂Ω) = 0.

This is a contraddiction, we conclude that µ_k+o() ≤ σ_k. Now recalling that σ_k ≤ µk+o() (see inequality (6)) we can conclude that

σ_k ∼µk+o().

We also conclude that

u_k(x1, x2)* Vk(x1, x2) in H¹(T1),

where V_k is the k−th eigenfunction of the problem (2) constantly extended to x₂. We end the proof by proving that the convergence is true not only up to a subsequence but is true for all the sequence. We have seen that the only possible accumulation point is Vk eigenfunction of Problem (18). Now it is a classical result for Sturm-Liouville type problem that any eigenfunction is simple: use the ODE to prove that the Wronskian is constant and the boundary conditions to prove that it is zero, yielding the result.

From the uniqueness of the accumulation point we conclude that the convergence holds for the whole sequence. This concludes the proof of Theorem 1.1

3. The casen≥3 and k≥2. Proof of Theorem 1.2.

In this section we will prove the second part of Theorem 1.2. We will use the following notation, takex∈Rⁿ then we writex= (x₁, x⁰) wherex₁∈Rand x⁰ ∈Rⁿ⁻¹

3.1. Upper bound for the Steklov eigenvalue. In this section we prove un upper bound for all the Steklov eigenvalues. In the following lemma we give an estimate from above of the speed of convergence to zero of the Steklov eigenvalues.

Lemma 3.1. Let n≥3 and let Ω⊂Rⁿ be a dumbbell shape domain then

• for the first Steklov eigenvalue

(19) σ₁≤σ₁ⁿ⁻¹+o(ⁿ⁻¹)

where σ₁ > 0 is the unique positive number such that the following differential equation has a non-trivial solution:











−w_n−1dx^d ρⁿ⁻¹(x)^dV_dx¹(x)

= 0 x∈ −^L₂,^L₂ ρⁿ⁻¹(−^L₂)^dV_dx¹(−^L₂) =−_ω^σ1

n−1P(D1)V1(−^L₂) ρⁿ⁻¹(^L₂)^dV_dx¹(^L₂) = _ω^σ1

n−1P(D₂)V₁(^L₂).

• For all the other Steklov eigenvalues (k≥2)

(20) σ_k≤λ_k+o()

where λ_k is defined by the following 1−dimensional eigenvalue problem:

(21)











−ω_n−1dx^d ρⁿ⁻¹(x)^dV_dx^k(x)

=λ_kωn−2ρⁿ⁻²(x)V_k(x) x∈ −^L₂,^L₂ ρⁿ⁻¹(−^L₂)^dV_dx^k(−^L₂) =−n−2^λk P(D₁)V_k(−^L₂)

ρⁿ⁻¹(^L₂)^dV_dx^k(^L₂) = n−2^λk P(D₂)V_k(^L₂).

Proof. We introduce the following functional space (22) H_co¹(Ω) =

u∈H¹(Ω)|u≡c_i inD_i Z

∂Ω

u= 0, andudepends only onx₁ inT and denote σ_k^co(Ω) the (pseudo) k-th Steklov eigenvalue computed by replacing the Sobolev space H¹(Ω) with H_co¹(Ω) in the variational formulation using the Rayleigh quotient. SinceH_co¹(Ω) is a subspace of H¹(Ω), we obtain:

σ₁ ≤σ₁^co(Ω) = inf

06=u∈H_co¹(Ω)

R

Ω|∇u|²dx R

∂Ωu²dHⁿ⁻¹

≤ inf

06=u∈H_co¹(Ω)

ⁿ⁻¹wn−1R^L₂

−^L₂ ρⁿ⁻¹(u⁰)²dx₁

P(D₁)u²(−^L₂) +P(D₂)u²(^L₂) +o(ⁿ⁻¹)

≤σ1ⁿ⁻¹+o(ⁿ⁻¹).

The last inequality is true because the quantity

06=u∈Hinf_co¹(Ω)

wn−1R^L₂

−^L₂ ρⁿ⁻¹(u⁰)²dx1

P(D1)u²(−^L₂) +P(D2)u²(^L₂)

is equal to σ₁ that is the unique positive number such that the following differential equation has a non-trivial solution:











−w_n−1dx^d ρⁿ⁻¹(x)^dV_dx¹(x)

= 0 x∈ −^L₂,^L₂ ρⁿ⁻¹(−^L₂)^dV_dx¹(−^L₂) =−_w^σ1

n−1P(D1)V1(−^L₂) ρⁿ⁻¹(^L₂)^dV_dx¹(^L₂) = _w^σ1

n−1P(D₂)V₁(^L₂).

We now prove the second part of the lemma. We start by noticing that from the geometric properties of∂T^e we can compute the surface measure and we obtain that

(23) dHⁿ⁻¹x∂T^e =ⁿ⁻²ρⁿ⁻²p

1 +²ρ⁰²dx1dϕ1...dϕn−2

LetS_k be the family of all the k−dimensional subspaces of the functional spaceH_co¹(Ω) withk≥2, as above we have the following inequalities

σ_k≤σ^co_k (Ω) = inf

E∈S_k+1sup

u∈E

R

Ω|∇u|²dx R

∂Ωu²dHⁿ⁻¹

≤ inf

E∈S_k+1sup

u∈E

ⁿ⁻¹wn−1

R^L₂

−^L

2

ρⁿ⁻¹(u⁰)²dx1

P(D₁)u²(−^L₂) +P(D₂)u²(^L₂) +ⁿ⁻²wn−2R^L₂

−^L₂ u²ρⁿ⁻²p

1 +²ρ⁰²dx₁ +o()

≤ inf

E∈S_k+1sup

u∈E

wn−1R^L₂

−^L₂ ρⁿ⁻¹(u⁰)²dx1 P(D1)

ⁿ⁻² u²(−^L₂) +^P(Dn−2²⁾u²(^L₂) +wn−2R^L₂

−^L₂ u²ρⁿ⁻²dx₁ +o()

≤λ_k+o().

Where the last inequality is true because the quantity wn−1R^L₂

−^L₂ ρⁿ⁻¹(u⁰)²dx₁

P(D1)

ⁿ⁻² u²(−^L₂) + ^P(Dn−2²⁾u²(^L₂) +wn−2

R^L₂

−^L

2

u²ρⁿ⁻²dx1

is the Rayleigh quotient of the following eigenvalue problem that depends on

(24)











−w_n−1dx^d ρⁿ⁻¹(x)^dV_dx^k(x)

=λ_kwn−2ρⁿ⁻²(x)Vk(x) x∈ − ^L₂,^L₂ ρⁿ⁻¹(−^L₂)^dV_dx^k(−^L₂) =−n−2^λk P(D1)Vk(−^L₂)

ρⁿ⁻¹(^L₂)^dV_dx^k(^L₂) = ^λ

k

ⁿ⁻²P(D2)Vk(^L₂).

3.2. Convergence of eigenfunctions. We begin with the convergence on the two regions D_i where i= 1,2. The proof of the following lemma is the same of the proof of Lemma 2.2, so we do not repeat it.

Lemma 3.2. Let k≥1 we have (up to a sub-sequence that we still denote by u_k) u_k* ci,k in H¹(Di),

u_k→c_i,k locally uniformly in D_i. where c_i,k are constants

We study the behaviour of the eigenfunctions in the tubeT. For everyk≥2 we define the following rescaled functions

v_k(x1, x⁰) =ⁿ⁻²² u_k(x1, x⁰) ∀(x1, x⁰)∈T1

Lemma 3.3. Let n≥3 and k≥2. There existsV_k∈H¹(T₁) which depends only on the variable x1 such that

v_k* V_k in H¹(T₁), up to a sub-sequence (that we still denote byv_k).

Proof. Below, by C we denote a constant which may change from line to line. We start with the bound of||∇v_k||_L2(T1)

Z

T1

|∇v_k|²dx≤ Z

T1

∂v_k

∂x1

2

+ 1

²|∇_x⁰v_k|²dx= 1

Z

T

|∇u_k|²dy≤C