Shape optimization for the eigenvalues of a biharmonic Steklov problem

Shape optimization for the eigenvalues of a biharmonic

Steklov problem

Luigi Provenzano

joint work with Davide Buoso PEPworkshop 2014, Aveiro November 06, 2014

The biharmonic Steklov problem

LetΩbe a bounded domain inR^{Nof class C1,}τ >0 a fixed constant.















∆²u−τ∆u=0, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν− div_∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω ,

2 of 26

The biharmonic Steklov problem

Z

Ω

D²u:D²φ + τ∇u· ∇φdx =λ Z

∂Ω

uφdσ , ∀φ ∈H²(Ω),

where D²u:D²φ = P^N_i,j=1∂x^∂2u

i∂xj

∂²φ

∂xi∂xj

0=λ₁[Ω]<λ₂[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

3 of 26

The biharmonic Steklov problem

Z

Ω

D²u:D²φ + τ∇u· ∇φdx =λ Z

∂Ω

uφdσ , ∀φ ∈H²(Ω),

where D²u:D²φ = P^N_i,j=1∂x^∂2u

i∂xj

∂²φ

∂xi∂xj

0=λ₁[Ω]<λ₂[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

3 of 26

The biharmonic Steklov problem

Ω 7→λ_j[Ω], Ω 7→λ₂[Ω]

max_Ωλ_j[Ω]? min_Ωλ_j[Ω] ? Critical points ? among setsΩwith a fixed volume|Ω|

4 of 26

The biharmonic Steklov problem

Ω 7→λ_j[Ω], Ω 7→λ₂[Ω]

max_Ωλ_j[Ω]? min_Ωλ_j[Ω] ? Critical points ? among setsΩwith a fixed volume|Ω|

4 of 26

Other Steklov-type problems

LetΩbe a bounded domain of class C¹inR^N















∆²u=0, in Ω , u=0, on ∂Ω ,

∆u=λ^∂u_∂ν, on ∂Ω ,

Bucur, Ferrero, Gazzola, “On the first eigenvalue of a fourth order Steklov problem”, Calc. Var. Partial Differential Equations, 35.

5 of 26

Other Steklov-type problems

LetΩbe a bounded domain of class C¹inR^N















∆²u=0, in Ω , u=0, on ∂Ω ,

∆u=λ^∂u_∂ν, on ∂Ω ,

Bucur, Ferrero, Gazzola, “On the first eigenvalue of a fourth order Steklov problem”, Calc. Var. Partial Differential Equations, 35.

5 of 26

Steklov problem for the Laplacian

Steklov problem for the Laplacian ( ∆u=0, in Ω ,

∂u∂ν =λu, on ∂Ω ,

0=λ₁[Ω]<λ₂[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume (Weinstock, Brock).

6 of 26

Steklov problem for the Laplacian

Steklov problem for the Laplacian ( ∆u=0, in Ω ,

∂u∂ν =λu, on ∂Ω ,

0=λ₁[Ω]<λ₂[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume (Weinstock, Brock).

6 of 26

Steklov problem for the Laplacian

Steklov problem for the Laplacian ( ∆u=0, in Ω ,

∂u∂ν =λu, on ∂Ω ,

0=λ₁[Ω]<λ₂[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume (Weinstock, Brock).

6 of 26

Neumann vs Steklov, second order

( −∆u=λ(ε)ρ_εu, in Ω ,

∂u∂ν =0, on ∂Ω ,

where

ρ_ε=

( ε, in Ω \ ¯ω_ε, C_ε, in ωε, ω_ε=

x∈Ω : dist(x, ∂Ω) < ε andR

Ωρ_ε=M for allε ∈]0, ε₀[.

7 of 26

Neumann vs Steklov, second order

( −∆u=λ(ε)ρ_εu, in Ω ,

∂u∂ν =0, on ∂Ω , where

ρ_ε=

( ε, in Ω \ ¯ω_ε, C_ε, in ωε, ωε=

x∈Ω : dist(x, ∂Ω) < ε andR

Ωρε=M for allε ∈]0, ε0[.

7 of 26

Neumann vs Steklov, second order

( −∆u=λ(ε)ρ_εu, in Ω ,

∂u∂ν =0, on ∂Ω , where

ρ_ε=

( ε, in Ω \ ¯ω_ε, C_ε, in ωε, ωε=

x∈Ω : dist(x, ∂Ω) < ε andR

Ωρε=M for allε ∈]0, ε0[.

7 of 26

Neumann vs Steklov, second order

For all j∈ N^,λ_j(ε) → λ_jasε →0

0.2 0.4 0.6 0.8

0 10 20 30 40

Figure: N=2, M=π

8 of 26

Neumann vs Steklov, second order

For all j∈ N^,λ_j(ε) → λ_jasε →0

0.2 0.4 0.6 0.8

0 10 20 30 40

Figure: N=2, M=π

8 of 26

The biharmonic Steklov problem

Strategy:

Biharmonic Neumann problem with mass densityρ_ε















∆²u−τ∆u=λ(ε)ρ_εu, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν− div∂Ω D²u.ν

− ^∂∆u_∂ν =0, on ∂Ω ,

Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume, whenρ_ε≡ const(Chasman 2011).

Write the HamiltonianH of a plate with its mass concentrated at the boundary and recover equations of motion

⇓















∆²u−τ∆u=0, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν − div∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω ,

9 of 26

The biharmonic Steklov problem

Strategy:

Biharmonic Neumann problem with mass densityρ_ε















∆²u−τ∆u=λ(ε)ρ_εu, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν− div∂Ω D²u.ν

− ^∂∆u_∂ν =0, on ∂Ω , Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume, whenρ_ε≡ const(Chasman 2011).

Write the HamiltonianH of a plate with its mass concentrated at the boundary and recover equations of motion

⇓















∆²u−τ∆u=0, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν − div∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω ,

9 of 26

The biharmonic Steklov problem

Strategy:

Biharmonic Neumann problem with mass densityρ_ε















∆²u−τ∆u=λ(ε)ρ_εu, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν− div∂Ω D²u.ν

− ^∂∆u_∂ν =0, on ∂Ω , Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume, whenρ_ε≡ const(Chasman 2011).

Write the HamiltonianH of a plate with its mass concentrated at the boundary and recover equations of motion

⇓















∆²u−τ∆u=0, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν − div∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω ,

9 of 26

The biharmonic Steklov problem

Strategy:

Biharmonic Neumann problem with mass densityρ_ε















∆²u−τ∆u=λ(ε)ρ_εu, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν− div∂Ω D²u.ν

− ^∂∆u_∂ν =0, on ∂Ω , Theballis a maximizer forλ₂[Ω]amongΩwith a fixed volume, whenρ_ε≡ const(Chasman 2011).

Write the HamiltonianH of a plate with its mass concentrated at the boundary and recover equations of motion

⇓















∆²u−τ∆u=0, in Ω ,

∂²u

∂ν² =0, on ∂Ω ,

τ^∂u_∂ν − div∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω ,

9 of 26

Symmetric functions of the eigenvalues

LetΩa bounded domain inR^{N. Set} Φ(Ω) =

φ ∈

C²(Ω)N

, injective : inf

Ω |detDφ| >0



Theorem (Buoso-P. 2014)

LetΩbe a bounded domain ofR^{Nof class C1}. Let F be a finite non-empty subset ofN \ {⁰}. Let

A_Ω[F] =nφ ∈ Φ(Ω) : λ_l[φ] <nλ_j[φ] :j∈Fo

∀l ∈ N \ (^F ∪ {0})o

Then the setA_Ωis open inΦ(Ω)and the mapΛ_F,sfromA_ΩtoR defined by

ΛF,s[φ] = X

j1<···<js∈F

λj1[φ] · · · λjs[φ] for s∈ {1, ..., |F|}is real analytic.

10 of 26

Symmetric functions of the eigenvalues

LetΩa bounded domain inR^{N. Set} Φ(Ω) =

φ ∈

C²(Ω)N

, injective : inf

Ω |detDφ| >0



Theorem (Buoso-P. 2014)

LetΩbe a bounded domain ofR^{N of class C1}. Let F be a finite non-empty subset ofN \ {⁰}. Let

A_Ω[F] =nφ ∈ Φ(Ω) : λ_l[φ] <nλ_j[φ] :j ∈Fo

∀l ∈ N \ (^F ∪ {0})o

Then the setA_Ω is open inΦ(Ω)and the mapΛ_F,sfromA_ΩtoR defined by

ΛF,s[φ] = X

j1<···<js∈F

λ_j1[φ] · · · λ_js[φ]

for s∈ {1, ..., |F|}is real analytic.

10 of 26

Isovolumetric perturbations

Theorem (Buoso-P. 2014)

LetΩbe a bounded domain inR^N. Let F a finite non-empty subset ofN \ {⁰}. Letφ ∈ A˜ _Ω[F]be such that all the eigenvalues with indexes in F have a commond valueλF and moreover that

∂˜φ(Ω) ∈C⁴. Let v₁, ...,v_|F|be a hortonormal basis of the eigenspace associated with the eigenvalueλ_F[˜φ]. Then

d|_φ=˜φ(Λ_F,s)[ψ] = −λ^s_F[˜φ] |F| −1 s−1

! |F|

X

l=1

Z

∂˜φ(Ω)

λ_FKv_l²

+λ_F∂(v_l²)

∂ν −τ|∇v_l|²− |D²v_l|²

µ · νdσ, (1.3)

for allψ ∈ (C²(Ω))^N, whereµ = ψ ◦ φ⁽⁻¹⁾, and K denotes the mean curvature on∂˜φ(Ω).

11 of 26

Isovolumetric perturbations

V(φ) = Z

φ(Ω)

dy = Z

Ω

|detDφ|dx FixV₀ ∈]0, +∞[

V(V₀) =φ ∈ Φ(Ω) : V(φ) = V₀

The functionφ˜is a critical point forΛ_F,sif and only if

|F|

X

l=1







λ_F[˜φ]







^Kv

2 l +∂v_l²

∂ν







−τ|∇v_l|²− |D²v_l|²







=c, a.e. on ∂˜φ(Ω).

Theorem (Buoso-P. 2014)

LetΩbe a domain ofR^{N. Let}φ ∈ Φ(Ω)˜ be such thatφ(Ω)˜ is a ball. Letλ˜be an eigenvalue of the problem inφ(Ω)˜ , and let F be the set of j∈ N \ {⁰}such thatλj[˜φ] = ˜λ. ThenΛF,s has a critical point atφ˜ on V(V(˜φ)), for all s =1, . . . , |F|.

12 of 26

Isovolumetric perturbations

V(φ) = Z

φ(Ω)

dy = Z

Ω

|detDφ|dx FixV₀ ∈]0, +∞[

V(V₀) =φ ∈ Φ(Ω) : V(φ) = V₀ The functionφ˜is a critical point forΛ_F,sif and only if

|F|

X

l=1







λ_F[˜φ]







^Kv

2 l +∂v_l²

∂ν







−τ|∇v_l|²− |D²v_l|²







=c, a.e. on ∂˜φ(Ω).

Theorem (Buoso-P. 2014)

LetΩbe a domain ofR^{N. Let}φ ∈ Φ(Ω)˜ be such thatφ(Ω)˜ is a ball. Letλ˜be an eigenvalue of the problem inφ(Ω)˜ , and let F be the set of j∈ N \ {⁰}such thatλj[˜φ] = ˜λ. ThenΛF,s has a critical point atφ˜ on V(V(˜φ)), for all s =1, . . . , |F|.

12 of 26

Isovolumetric perturbations

V(φ) = Z

φ(Ω)

dy = Z

Ω

|detDφ|dx FixV₀ ∈]0, +∞[

V(V₀) =φ ∈ Φ(Ω) : V(φ) = V₀ The functionφ˜is a critical point forΛ_F,sif and only if

|F|

X

l=1







λ_F[˜φ]







^Kv

2 l +∂v_l²

∂ν







−τ|∇v_l|²− |D²v_l|²







=c, a.e. on ∂˜φ(Ω).

Theorem (Buoso-P. 2014)

LetΩbe a domain ofR^{N. Let}φ ∈ Φ(Ω)˜ be such thatφ(Ω)˜ is a ball.

Letλ˜be an eigenvalue of the problem inφ(Ω)˜ , and let F be the set of j∈ N \ {⁰}such thatλ_j[˜φ] = ˜λ. ThenΛF,s has a critical point atφ˜ on V(V(˜φ)), for all s =1, . . . , |F|.

12 of 26

The fundamental tone

Balls are critical for the symmetric functions of the eigenvalues under isovolumetric perturbations

Could we say more on the fundamental toneλ₂?

Theorem (Buoso-P. 2014)

Among all bounded domains of class C¹ with fixed volume, the ball maximizes the first non-negative eigenvalue, that isλ₂[Ω] ≤λ₂[Ω^∗], whereΩ^∗is the ball with the same volume asΩ.

13 of 26

The fundamental tone

Balls are critical for the symmetric functions of the eigenvalues under isovolumetric perturbations

Could we say more on the fundamental toneλ₂?

Theorem (Buoso-P. 2014)

Among all bounded domains of class C¹ with fixed volume, the ball maximizes the first non-negative eigenvalue, that isλ₂[Ω] ≤λ₂[Ω^∗], whereΩ^∗is the ball with the same volume asΩ.

13 of 26

The fundamental tone

Balls are critical for the symmetric functions of the eigenvalues under isovolumetric perturbations

Could we say more on the fundamental toneλ₂?

Theorem (Buoso-P. 2014)

Among all bounded domains of class C¹ with fixed volume, the ball maximizes the first non-negative eigenvalue, that isλ₂[Ω] ≤λ₂[Ω^∗], whereΩ^∗is the ball with the same volume asΩ.

13 of 26

The fundamental tone

ConsiderB =B(0,1)⊂ R^N.

All the eigenfunctions of the Steklov problem are of the form

u(r, θ₁, ..., θ_N−1) =R_l(r)Y_l(θ₁, ..., θ_N−1) where

R_l(r) =α_lr^l+β_li_l(√ τr).

The corresponding eigenvalues are given by an explicit formula (rather long)

λ =g(l,N, τ), for l∈ N^.

Example: g(0,N, τ) =0, g(1,N, τ) = τ. Which l∈ N^{gives the} fundamental tone?

14 of 26

The fundamental tone

ConsiderB =B(0,1)⊂ R^N. All the eigenfunctions of the Steklov problem are of the form

u(r, θ₁, ..., θ_N−1) =R_l(r)Y_l(θ₁, ..., θ_N−1) where

R_l(r) =α_lr^l+β_li_l(√ τr).

The corresponding eigenvalues are given by an explicit formula (rather long)

λ =g(l,N, τ), for l∈ N^.

Example: g(0,N, τ) =0, g(1,N, τ) = τ. Which l∈ N^{gives the} fundamental tone?

14 of 26

The fundamental tone

ConsiderB =B(0,1)⊂ R^N. All the eigenfunctions of the Steklov problem are of the form

u(r, θ₁, ..., θ_N−1) =R_l(r)Y_l(θ₁, ..., θ_N−1) where

R_l(r) =α_lr^l+β_li_l(√ τr).

The corresponding eigenvalues are given by an explicit formula (rather long)

λ =g(l,N, τ), for l∈ N^.

Example: g(0,N, τ) =0, g(1,N, τ) = τ. Which l∈ N^{gives the} fundamental tone?

14 of 26

The fundamental tone

ConsiderB =B(0,1)⊂ R^N. All the eigenfunctions of the Steklov problem are of the form

u(r, θ₁, ..., θ_N−1) =R_l(r)Y_l(θ₁, ..., θ_N−1) where

R_l(r) =α_lr^l+β_li_l(√ τr).

The corresponding eigenvalues are given by an explicit formula (rather long)

λ =g(l,N, τ), for l∈ N^.

Example: g(0,N, τ) =0, g(1,N, τ) = τ. Which l∈ N^{gives the} fundamental tone?

14 of 26

The fundamental tone

λ₂[B]=g(1,N, τ) =τ

λ₂[B]has multiplicity N and the eigenfunctions are{x₁, ...,x_N}

Strategy: use the eigenfunctions of the unit ball as test functions in a variational characterization ofλ₂[Ω]

15 of 26

The fundamental tone

λ₂[B]=g(1,N, τ) =τ

λ₂[B]has multiplicity N and the eigenfunctions are{x₁, ...,x_N}

Strategy: use the eigenfunctions of the unit ball as test functions in a variational characterization ofλ₂[Ω]

15 of 26

The fundamental tone

λ₂[B]=g(1,N, τ) =τ

λ₂[B]has multiplicity N and the eigenfunctions are{x₁, ...,x_N}

Strategy: use the eigenfunctions of the unit ball as test functions in a variational characterization ofλ₂[Ω]

15 of 26

The fundamental tone

Lemma (Hile-Xu 1993)

LetΩbe a bounded domain of class C¹inR^{N. Then}

N+1

X

l=2

1

λ_l(Ω) =max











N+1

X

l=2

Z

∂Ωv_l²dσ









 ,

where{v_l}^N+1_l=2 is a family in H²(Ω)satisfying R

ΩD²v_i:D²v_j+τ∇v_i· ∇v_jdx =δij andR

∂Ωv_ldσ =0 for all l=2, ...,N+1.

16 of 26

The fundamental tone

Lemma (Betta-Brock-Mercaldo-Posteraro 1999)

LetΩbe an open set inR^N and f be a continuous, non-negative, non-decreasing function defined on[0, +∞). Let us assume that the function

t 7→

f(t^1/N) −f(0) t^1−(1/N) is convex. Then

Z

∂Ωf(|x|)dσ ≥ Z

∂Ω^∗f(|x|)dσ,

whereΩ^∗is the ball centered at zero with the same volume asΩ.

17 of 26

The fundamental tone

TakeΩof class C¹with|Ω| = |B|.

Perform the translation x_i =y_i−t_i

t_i = ¹

|∂Ω| Z

∂Ωy_idσ

Use test functionsv_l =(τ|Ω|)⁻¹²x_lin the variational formula and use the isoperimetric inequality

N+1X

l=2

1

λl[Ω] ≥ ¹ τ|Ω|

Z

∂Ω|x|²dσ ≥ ¹ τ|Ω|

Z

∂B|x|²dσ = ^N τ =

N+1X

l=2

1 λl[B].

Remark: for general values of|Ω|just observe λ[τ, Ω] =s⁴λ[s⁻²τ,sΩ]

18 of 26

The fundamental tone

TakeΩof class C¹with|Ω| = |B|. Perform the translation x_i =y_i−t_i

t_i = ¹

|∂Ω|

Z

∂Ωy_idσ

Use test functionsv_l =(τ|Ω|)⁻¹²x_lin the variational formula and use the isoperimetric inequality

N+1X

l=2

1

λl[Ω] ≥ ¹ τ|Ω|

Z

∂Ω|x|²dσ ≥ ¹ τ|Ω|

Z

∂B|x|²dσ = ^N τ =

N+1X

l=2

1 λl[B].

Remark: for general values of|Ω|just observe λ[τ, Ω] =s⁴λ[s⁻²τ,sΩ]

18 of 26

The fundamental tone

TakeΩof class C¹with|Ω| = |B|. Perform the translation x_i =y_i−t_i

t_i = ¹

|∂Ω|

Z

∂Ωy_idσ

Use test functionsv_l =(τ|Ω|)⁻¹²x_lin the variational formula and use the isoperimetric inequality

N+1X

l=2

1

λl[Ω] ≥ ¹ τ|Ω|

Z

∂Ω|x|²dσ ≥ ¹ τ|Ω|

Z

∂B|x|²dσ = ^N τ =

N+1X

l=2

1 λl[B].

Remark: for general values of|Ω|just observe λ[τ, Ω] =s⁴λ[s⁻²τ,sΩ]

18 of 26

The fundamental tone

TakeΩof class C¹with|Ω| = |B|. Perform the translation x_i =y_i−t_i

t_i = ¹

|∂Ω|

Z

∂Ωy_idσ

Use test functionsv_l =(τ|Ω|)⁻¹²x_lin the variational formula and use the isoperimetric inequality

N+1X

l=2

1

λ_l[Ω] ≥ ¹ τ|Ω|

Z

∂Ω|x|²dσ ≥ ¹ τ|Ω|

Z

∂B|x|²dσ = ^N τ =

N+1X

l=2

1 λ_l[B].

Remark: for general values of|Ω|just observe λ[τ, Ω] =s⁴λ[s⁻²τ,sΩ]

18 of 26

The fundamental tone

TakeΩof class C¹with|Ω| = |B|. Perform the translation x_i =y_i−t_i

t_i = ¹

|∂Ω|

Z

∂Ωy_idσ

Use test functionsv_l =(τ|Ω|)⁻¹²x_lin the variational formula and use the isoperimetric inequality

N+1X

l=2

1

λ_l[Ω] ≥ ¹ τ|Ω|

Z

∂Ω|x|²dσ ≥ ¹ τ|Ω|

Z

∂B|x|²dσ = ^N τ =

N+1X

l=2

1 λ_l[B].

Remark: for general values of|Ω|just observe λ[τ, Ω] =s⁴λ[s⁻²τ,sΩ]

18 of 26

Further directions: the case τ = 0

Letτ =0 andΩbe a bounded domain of class C¹















∆²u=0, in Ω,

∂²u

∂ν² =0, on ∂Ω,

−div_∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω,

0=λ₁[Ω] =λ₂[Ω] = · · · =λ_N+1[Ω]<λ_N+2[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

The kernel is{1,x₁,...,x_N}

19 of 26

Further directions: the case τ = 0

Letτ =0 andΩbe a bounded domain of class C¹















∆²u=0, in Ω,

∂²u

∂ν² =0, on ∂Ω,

−div_∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω,

0=λ₁[Ω] =λ₂[Ω] = · · · =λ_N+1[Ω]<λ_N+2[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

The kernel is{1,x₁,...,x_N}

19 of 26

Further directions: the case τ = 0

Letτ =0 andΩbe a bounded domain of class C¹















∆²u=0, in Ω,

∂²u

∂ν² =0, on ∂Ω,

−div_∂Ω D²u.ν

− ^∂∆u_∂ν =λu, on ∂Ω,

0=λ₁[Ω] =λ₂[Ω] = · · · =λ_N+1[Ω]<λ_N+2[Ω]≤ · · · ≤λ_j[Ω] ≤ · · ·

The kernel is{1,x₁,...,x_N}

19 of 26

Further directions: the case τ = 0

What we can do:

identify the fundamental tone of the unit ball λ_N+2[B]=2 N+⁸

5

!

identify the corresponding eigenfunctions

u(r, θ₁, ..., θ_N−1) =(6r²−r⁴)Y₂(θ₁, ..., θ_N−1)

construct trial functions of the formR(r)Y₂(θ₁, ...θ_N+1)

20 of 26

Further directions: the case τ = 0

What we can do:

identify the fundamental tone of the unit ball λ_N+2[B]=2 N+⁸

5

!

identify the corresponding eigenfunctions

u(r, θ₁, ..., θ_N−1) =(6r²−r⁴)Y₂(θ₁, ..., θ_N−1)

construct trial functions of the formR(r)Y₂(θ₁, ...θ_N+1)

20 of 26

Further directions: the case τ = 0

What we can do:

identify the fundamental tone of the unit ball λ_N+2[B]=2 N+⁸

5

!

identify the corresponding eigenfunctions

u(r, θ₁, ..., θ_N−1) =(6r²−r⁴)Y₂(θ₁, ..., θ_N−1)

construct trial functions of the formR(r)Y₂(θ₁, ...θ_N+1)

20 of 26

Further directions: the case τ = 0

What we can do:

identify the fundamental tone of the unit ball λ_N+2[B]=2 N+⁸

5

!

identify the corresponding eigenfunctions

u(r, θ₁, ..., θ_N−1) =(6r²−r⁴)Y₂(θ₁, ..., θ_N−1)

construct trial functions of the formR(r)Y₂(θ₁, ...θ_N+1)

20 of 26

Further directions: the case τ = 0

What we (still) cannot do:

test these trial functions onanyΩof class C¹

find good estimates for the sum of the reciprocals in the case the test is possible

Trial functions work withradial domains. For small dimensions we haveisoperimetric inequality

Theorem (Buoso-P. 2014)

Among all bounded radial domainsΩwith a fixed volume inR^N, N≤4, the ball maximizes the first non-zero eigenvalue, that is

λ_N+2[Ω] ≤λ_N+2[Ω^∗], whereΩ^∗is the ball with the same volume ofΩ.

21 of 26

Further directions: the case τ = 0

What we (still) cannot do:

test these trial functions onanyΩof class C¹

find good estimates for the sum of the reciprocals in the case the test is possible

Trial functions work withradial domains. For small dimensions we haveisoperimetric inequality

Theorem (Buoso-P. 2014)

Among all bounded radial domainsΩwith a fixed volume inR^N, N≤4, the ball maximizes the first non-zero eigenvalue, that is

λ_N+2[Ω] ≤λ_N+2[Ω^∗], whereΩ^∗is the ball with the same volume ofΩ.

21 of 26

Further directions: the case τ = 0

What we (still) cannot do:

test these trial functions onanyΩof class C¹

find good estimates for the sum of the reciprocals in the case the test is possible

Trial functions work withradial domains. For small dimensions we haveisoperimetric inequality

Theorem (Buoso-P. 2014)

Among all bounded radial domainsΩwith a fixed volume inR^N, N≤4, the ball maximizes the first non-zero eigenvalue, that is

λ_N+2[Ω] ≤λ_N+2[Ω^∗], whereΩ^∗is the ball with the same volume ofΩ.

21 of 26

Further directions: the case τ = 0

What we (still) cannot do:

test these trial functions onanyΩof class C¹

find good estimates for the sum of the reciprocals in the case the test is possible

Trial functions work withradial domains. For small dimensions we haveisoperimetric inequality

Theorem (Buoso-P. 2014)

Among all bounded radial domainsΩwith a fixed volume inR^N, N≤4, the ball maximizes the first non-zero eigenvalue, that is

λ_N+2[Ω] ≤λ_N+2[Ω^∗], whereΩ^∗is the ball with the same volume ofΩ.

21 of 26

Further directions: the case τ = 0

N=2 N=3 N=4 N=5

0.5 1 1.5 R

0 1 2 3 4 5 6 36 5 46 5 56 5 66 5 Λ_2,1HRL

Figure:N=2,3,4,5

22 of 26

Further directions: Neumann problem, Poly-harmonic operators,...

Neumann problem for the biharmonic operator















∆²u=λu, in Ω,

∂²u

∂ν² =0, on ∂Ω,

−div∂Ω D²u.ν

− ^∂(∆u)_∂ν =0, on ∂Ω.

Λ0,1 Λ1,1 Λ_2,1

10 20 30 40 50 60 Λ

50 100 150 200

Figure:N=2

23 of 26

Further directions: Neumann problem, Poly-harmonic operators,...

Neumann problem for the biharmonic operator















∆²u=λu, in Ω,

∂²u

∂ν² =0, on ∂Ω,

−div∂Ω D²u.ν

− ^∂(∆u)_∂ν =0, on ∂Ω.

Λ0,1 Λ_1,1 Λ_2,1

10 20 30 40 50 60 Λ

50 100 150 200

Figure:N=2

23 of 26

Further directions: Neumann problem, Poly-harmonic operators,...

Neumann problem for(−∆)^m

( (−∆)^mu=λu, in Ω,

N₁u=N₂u= · · · =N_mu=0, on ∂Ω,

N_iu are the m natural boundary conditions, ordered according their order: N₁ is an operator of order m, N₂ is of order m+1,..., N_m is of order 2m−1.

Steklov problem for(−∆)^m











∆^mu=0, in Ω,

N₁u=N₂u= · · · =N_m−1u=0, on ∂Ω,

N_mu=λu, on ∂Ω,

with the same N_i

24 of 26

Further directions: Neumann problem, Poly-harmonic operators,...

Neumann problem for(−∆)^m

( (−∆)^mu=λu, in Ω,

N₁u=N₂u= · · · =N_mu=0, on ∂Ω,

N_iu are the m natural boundary conditions, ordered according their order: N₁ is an operator of order m, N₂ is of order m+1,..., N_m is of order 2m−1.

Steklov problem for(−∆)^m











∆^mu=0, in Ω,

N₁u=N₂u= · · · =N_m−1u=0, on ∂Ω,

N_mu=λu, on ∂Ω,

with the same N_i

24 of 26

References

C. Bandle,

Isoperimetric inequalities and applications,

Pitman advanced publishing program, monographs and studies in mathematics, vol. 7, 1980.

D. Bucur, A. Ferrero, F. Gazzola,

On the first eigenvalue of a fourth order Steklov problem,

Calculus of Variations and Partial Differential Equations, 35 , 103-131, 2009.

D. Buoso, L. Provenzano,

An isoperimetric inequality for the first non-zero eigenvalue of a biharmonic Steklov problem.

preprint, 2014.

D. Gomez, M. Lobo, E. Perez,

On the vibrations of a plate with a concentrated mass and very small thickness, Math. Method. Appl. Sci. 26, no.2, 27-65, 2003.

D. Gomez, M. Lobo, S.A. Nazarov, E. Perez,

Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl. 85, no.4, 598-632, 2006.

L.M. Chasman,

An isoperimetric inequality for fundamental tones of free plates, Comm. Math. Phys. 303, no. 2, 421-449, 2011.

P.D. Lamberti, L. Provenzano,

Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues, to appear on the 9th Isaac Congress proceedings.

25 of 26

THANK YOU

26 of 26

Further directions: mass concentration

Behavior ofλ_j(ε)for mass concentration problem for the biharmonic operator

0.2 0.4 0.6 0.8

0 20 40 60 80 100

Figure:N=2, M=π, τ = 5

On the ball? On arbitraryΩ(also in the second order case)?

26 of 26

Further directions: mass concentration

Behavior ofλ_j(ε)for mass concentration problem for the biharmonic operator

0.2 0.4 0.6 0.8

0 20 40 60 80 100

Figure:N=2, M=π, τ = 5

On the ball? On arbitraryΩ(also in the second order case)?

26 of 26

Further directions: mass concentration

Behavior ofλ_j(ε)for mass concentration problem for the biharmonic operator

0.2 0.4 0.6 0.8

0 20 40 60 80 100

Figure:N=2, M=π, τ = 5

On the ball? On arbitraryΩ(also in the second order case)?

26 of 26