Spectral analysis of a Neumannbiharmonic operator on a planar

dumbbell domain

Francesco Ferraresso(based on a joint work with J.M. Arrieta and P.D. Lamberti)

Workshop on geometric spectral theoryNeuchatel - 22.06.2017

The dumbbell

Ω

ΩRε

1

(a) Dumbbell domain.

Ω

ΩR0

1

(b) Limit domain.

2 of 19

The dumbbell

Ω

ΩRε

1

(a) Dumbbell domain.

Ω

ΩR0

1

(b) Limit domain.

2 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Dumbbells in the literature

Spectral analysis on dumbbells is important in:

Geometric spectral theory→ Cheeger’s constant;

Parabolic reaction-diffusion equations→ Matano’scounterexample;

Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.

Many contributions in the case of the Laplace operator:

Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...

Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...

...and many others.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN.

Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω)

with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent,

itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete

λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε

in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

Motivations: eigenvalues and perturbations

Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .

Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.

There are many cases of quite regular perturbations providingcounterexamples

Aim: better understanding of this “pathological” cases.

3 of 19

The dumbbell problem

The dumbbell domains Ωε ⊂ R2 are perturbations of a fixed

bounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

The dumbbell problemThe dumbbell domains Ωε ⊂ R

2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

The dumbbell problemThe dumbbell domains Ωε ⊂ R

2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR

Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

The dumbbell problemThe dumbbell domains Ωε ⊂ R

2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

The dumbbell problemThe dumbbell domains Ωε ⊂ R

2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

The dumbbell problemThe dumbbell domains Ωε ⊂ R

2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.

Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,

Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).

x

y

0 1

εg(0) Rεεg(1)

ΩL

ΩR

1

4 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.

5 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.

5 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε).

The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.

5 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis:

find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.

5 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.

5 of 19

A dumbbell-shaped free plate

On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2

∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,

τ∂u∂n − (1 − σ) div∂Ωε (D

2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .

where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds

∫Ωε

(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)

∫Ωε

uψ dx

We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.5 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0

if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2

i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ]

and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral convergence

Definition

Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if

λn[Ωε ]→ λn[Ω0] for all n ∈ N

The spectral projections PΩεa converge to PΩ0

a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the

projections PΩεa from L2(RN) into L2(Ωε) by

PΩεa (ψ) =

∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that

sup

∥∥∥∥PΩεa (ψ) − PΩ0

a (ψ)∥∥∥∥

L2(Ω0)+

∥∥∥PΩεa (ψ)

∥∥∥L2(Ωε\Ω0)

→ 0,

where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.

6 of 19

Spectral stability

We say that there is spectral stability

if the operator Tε := ∆2Ωε

associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,

(NBC)σ, on ∂Ω

7 of 19

Spectral stability

We say that there is spectral stability if the operator Tε := ∆2Ωε

associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,

(NBC)σ, on ∂Ω

7 of 19

Spectral stability

We say that there is spectral stability if the operator Tε := ∆2Ωε

associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

spectrally converges to the operator T0 := ∆2Ω

associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,

(NBC)σ, on ∂Ω

7 of 19

Spectral stability

We say that there is spectral stability if the operator Tε := ∆2Ωε

associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,

(NBC)σ, on ∂Ω

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Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:

There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Characterization of the spectral stability

General condition for bounded open sets Ωε , Ω.

Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that

|Ω \ Kε | → 0, as ε → 0

If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then

limε→0‖vε‖L2(Ωε\Kε) = 0

Arrieta’s condition⇒ Spectral convergence

8 of 19

Spectral stability II

In our case, Arrieta’s condition is related to the limit as ε → 0 of

τε = infφε ∈W2,2(Ωε)φε=0 in Ω

∫Ωε

(1 − σ)∣∣∣D2φε

∣∣∣2 + σ |∆φε |2 + τ |∇φε |

2 + |φε |2 dx

‖φε‖2L2(Rε)

Arrieta’s condition is equivalent to limε→0 τε = +∞.

The dumbbell Ωε violates this condition!

9 of 19

Spectral stability II

In our case, Arrieta’s condition is related to the limit as ε → 0 of

τε = infφε ∈W2,2(Ωε)φε=0 in Ω

∫Ωε

(1 − σ)∣∣∣D2φε

∣∣∣2 + σ |∆φε |2 + τ |∇φε |

2 + |φε |2 dx

‖φε‖2L2(Rε)

Arrieta’s condition is equivalent to limε→0 τε = +∞.

The dumbbell Ωε violates this condition!

9 of 19

Spectral stability II

In our case, Arrieta’s condition is related to the limit as ε → 0 of

τε = infφε ∈W2,2(Ωε)φε=0 in Ω

∫Ωε

(1 − σ)∣∣∣D2φε

∣∣∣2 + σ |∆φε |2 + τ |∇φε |

2 + |φε |2 dx

‖φε‖2L2(Rε)

Arrieta’s condition is equivalent to limε→0 τε = +∞.

The dumbbell Ωε violates this condition!

9 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon:

it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Spectral instability

Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a

Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:

‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1

or equivalently, τε < ∞.

Concentration phenomenon⇒ Spectral instability

There must be extra eigenvalues in the limit!

10 of 19

Main problem

Question: can we characterize these extra eigenvalues?

10 of 19

Two auxiliary problems

We introduce the eigenpairs (ϕΩk , ωk ) of∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω,

and the eigenpairs (γεl , θεl ) of

∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

where Γε = (x, y) : 0 < x < 1, y = εg(x) ∪ (x, 0) : 0 < x < 1and Lε = ∂Rε \ Γε .

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Two auxiliary problems

We introduce the eigenpairs (ϕΩk , ωk ) of∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω,

and the eigenpairs (γεl , θεl ) of

∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

where Γε = (x, y) : 0 < x < 1, y = εg(x) ∪ (x, 0) : 0 < x < 1and Lε = ∂Rε \ Γε .

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x

y

0

Γε

Lε

1

εg(0) εg(1)

1

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An additional assumption

Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω.

We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:

x

y

0 1

εg(0) εg(1)

1

12 of 19

An additional assumption

Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.

This condition is usually called H-Condition:

x

y

0 1

εg(0) εg(1)

1

12 of 19

An additional assumption

Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:

x

y

0 1

εg(0) εg(1)

1

12 of 19

An additional assumption

Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:

x

y

0 1

εg(0) εg(1)

1

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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

and that ωk and θεl are the eigenvalues of

∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω

∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε

v = 0 = ∂v∂n , on Lε

Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then

|λn(Ωε) − λεn | → 0, as ε → 0.

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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

and that ωk and θεl are the eigenvalues of

∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω

∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε

v = 0 = ∂v∂n , on Lε

Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then

|λn(Ωε) − λεn | → 0, as ε → 0.

13 of 19

Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

and that ωk and θεl are the eigenvalues of

∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω

∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε

v = 0 = ∂v∂n , on Lε

Define (λεn)n = (ωk )k ∪ (θεl )l .

Assume that Rε satisfies theH-condition. Then

|λn(Ωε) − λεn | → 0, as ε → 0.

13 of 19

Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

and that ωk and θεl are the eigenvalues of

∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω

∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε

v = 0 = ∂v∂n , on Lε

Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition.

Then

|λn(Ωε) − λεn | → 0, as ε → 0.

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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,

(NBC)σ, on ∂Ωε

and that ωk and θεl are the eigenvalues of

∆2w − τ∆w + w = ωk w, in Ω

(NBC)σ, on ∂Ω

∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε

v = 0 = ∂v∂n , on Lε

Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then

|λn(Ωε) − λεn | → 0, as ε → 0.

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A difficult problem

At this point we are interested in the convergence of θεl as ε → 0.

We would like to pass directly to the limit in the PDE problem∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

but this seems a tough problem. We took a different path.

14 of 19

A difficult problem

At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem

∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

but this seems a tough problem. We took a different path.

14 of 19

A difficult problem

At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem

∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

but this seems a tough problem.

We took a different path.

14 of 19

A difficult problem

At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem

∆2v − τ∆v + v = θεl v , in Rε

(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε

τ∂v∂n − (1 − σ) divΓε (D

2v · n)Γε −∂(∆v)∂n = 0, on Γε ,

v = 0 = ∂v∂n , on Lε .

but this seems a tough problem. We took a different path.

14 of 19

Compact convergence approach

Proof of the convergence to a suitable one-dimensional problemvia:

Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;

Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];

Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;

Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.

15 of 19

Compact convergence approach

Proof of the convergence to a suitable one-dimensional problemvia:

Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;

Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];

Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;

Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.

15 of 19

Compact convergence approach

Proof of the convergence to a suitable one-dimensional problemvia:

Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;

Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];

Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;

Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.

15 of 19

Compact convergence approach

Proof of the convergence to a suitable one-dimensional problemvia:

Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;

Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];

Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;

Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.

15 of 19

Compact convergence approach

Proof of the convergence to a suitable one-dimensional problemvia:

Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;

Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];

Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;

Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.

15 of 19

The limit problem

One can prove that∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε ,

v = 0 = ∂v∂n , on Lε .

compact converges to a suitable rescaling of1−σ2

g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)

v(0) = v(1) = 0, v′(0) = v′(1) = 0

We denote by(hi , θi) the eigenpairs associated with this ODE.

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The limit problem

One can prove that∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε ,

v = 0 = ∂v∂n , on Lε .

compact converges to a suitable rescaling of1−σ2

g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)

v(0) = v(1) = 0, v′(0) = v′(1) = 0

We denote by(hi , θi) the eigenpairs associated with this ODE.

16 of 19

The limit problem

One can prove that∆2v − τ∆v + v = θεl v , in Rε

(NBC)σ, on Γε ,

v = 0 = ∂v∂n , on Lε .

compact converges to a suitable rescaling of1−σ2

g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)

v(0) = v(1) = 0, v′(0) = v′(1) = 0

We denote by(hi , θi) the eigenpairs associated with this ODE.

16 of 19

Notation

Define the operator Eε : H2(0, 1)→ H2(Rε) by

Eεv(x, y) = v(x)

for all (x, y) ∈ Rε . Moreover let N(·) be the counting functiondefined by

N(x) = #λi : i ∈ N, λi ≤ x

17 of 19

Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk

or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl .

Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1,

and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0.

Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω)

and∥∥∥∥∥∥∥∥ϕεn −N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.

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Final convergence result

Theorem (Arrieta, F., Lamberti, 2016)

Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.

The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then

∥∥∥ϕεn |Ω∥∥∥L2(Ω)

→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑

i=1

(ϕεn, ϕΩi )L2(Ω)ϕ

Ωi

∥∥∥∥∥∥∥∥H2(Ω)

→ 0

as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −

N(θl)∑i=1

(ϕεn, ε−1/2Eεhi)L2(Rε)ε

−1/2Eεhi

∥∥∥∥∥∥∥∥L2(Rε)

→ 0

as ε → 0.18 of 19

Principal references

S. Jimbo, “The singularly perturbed domain and the characterizationfor the eigenfunctions with Neumann boundary conditions”,J.Differential Equations, 77, 1989, 322-350.

J. M. Arrieta, “Neumann eigenvalue problems on exteriorperturbations of the domain”, J. Differential Equations, 117, 1995.

J. M. Arrieta, A. N. Carvalho, G. Losada-Cruz, “Dynamics indumbbell domains I. Continuity of the set of equilibria”,J. Differential Equations, 231 (2), 2006, 551-597.

J. M. Arrieta, F.F., P.D. Lamberti, “Spectral analysis of thebiharmonic operator subject to Neumann boundary conditions ondumbbell domains”, submitted, 2017

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Thank youfor your attention

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Open questions

1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .

2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.

Partial results are available in the case σ = 0, the case σ , 0being open at the moment.

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Open questions

1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .

2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.

Partial results are available in the case σ = 0, the case σ , 0being open at the moment.

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Open questions

1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .

2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.

Partial results are available in the case σ = 0, the case σ , 0being open at the moment.

19 of 19