Spectral stability and boundary homogenization for polyharmonic - PowerPoint PPT Presentation

Spectral stability and boundary homogenization for polyharmonic operators Francesco Ferraresso Joint work with Pier Domenico Lamberti Kalamata 31.8.2015

Principal references J. A rrieta , P .D.L amberti , Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems, preprint, online at arXiv:1502.04373v2 [math.AP] F.F ., P .D.L amberti , Spectral convergence of higher order operators on varying domains and polyharmonic boundary homogenization, in preparation. 2 of 18

The general spectral problem 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m subject to homogeneous boundary conditions 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m subject to homogeneous boundary conditions (Dirichlet, Neumann, Intermediate etc.) 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m subject to homogeneous boundary conditions (Dirichlet, Neumann, Intermediate etc.) Under suitable assumptions, the spectrum is discrete λ 1 [Ω] ≤ λ 2 [Ω] ≤ · · · ≤ λ n [Ω] ≤ . . . 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m subject to homogeneous boundary conditions (Dirichlet, Neumann, Intermediate etc.) Under suitable assumptions, the spectrum is discrete λ 1 [Ω] ≤ λ 2 [Ω] ≤ · · · ≤ λ n [Ω] ≤ . . . Consider the functions Ω �→ λ n [Ω] , Ω → u n [Ω] 3 of 18

The general spectral problem Let Ω be a bounded open set in R N . We consider elliptic operators of the type � Hu = ( − 1 ) m D α � A αβ ( x ) D β u � x ∈ Ω , , | α | = | β | = m subject to homogeneous boundary conditions (Dirichlet, Neumann, Intermediate etc.) Under suitable assumptions, the spectrum is discrete λ 1 [Ω] ≤ λ 2 [Ω] ≤ · · · ≤ λ n [Ω] ≤ . . . Consider the functions Ω �→ λ n [Ω] , Ω → u n [Ω] Are they continuous? 3 of 18

Example: the bi-harmonic operator (vibrating plate) 4 of 18

Example: the bi-harmonic operator (vibrating plate) Dirichlet boundary conditions (clamped plate) 4 of 18

Example: the bi-harmonic operator (vibrating plate) Dirichlet boundary conditions (clamped plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∂ u  ∂ n = 0 , on ∂ Ω .   4 of 18

Example: the bi-harmonic operator (vibrating plate) Dirichlet boundary conditions (clamped plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∂ u  ∂ n = 0 , on ∂ Ω .   Neumann boundary conditions (free plate) 4 of 18

Example: the bi-harmonic operator (vibrating plate) Dirichlet boundary conditions (clamped plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∂ u  ∂ n = 0 , on ∂ Ω .   Neumann boundary conditions (free plate) ∆ 2 u = λ u ,  in Ω ,     ν ∆ u + ( 1 − ν ) ∂ 2 u  ∂ n 2 = 0 , on ∂ Ω ,     ( 1 − ν ) div ∂ Ω ( Hu · n ) + ∂ ∆ u  ∂ n = 0 , on ∂ Ω ,   4 of 18

Example: the bi-harmonic operator (vibrating plate) Dirichlet boundary conditions (clamped plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∂ u  ∂ n = 0 , on ∂ Ω .   Neumann boundary conditions (free plate) ∆ 2 u = λ u ,  in Ω ,     ν ∆ u + ( 1 − ν ) ∂ 2 u  ∂ n 2 = 0 , on ∂ Ω ,     ( 1 − ν ) div ∂ Ω ( Hu · n ) + ∂ ∆ u  ∂ n = 0 , on ∂ Ω ,   ν is the Poisson coefficient of the material (0 < ν < 1 / 2). 4 of 18

Example: the bi-harmonic operator Intermediate boundary conditions (hinged plate) 5 of 18

Example: the bi-harmonic operator Intermediate boundary conditions (hinged plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∆ u − k ( x ) ∂ u  ∂ n = 0 , on ∂ Ω   5 of 18

Example: the bi-harmonic operator Intermediate boundary conditions (hinged plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∆ u − k ( x ) ∂ u  ∂ n = 0 , on ∂ Ω   tricky case... 5 of 18

Example: the bi-harmonic operator Intermediate boundary conditions (hinged plate)  ∆ 2 u = λ u , in Ω ,      u = 0 , on ∂ Ω ,     ∆ u − k ( x ) ∂ u  ∂ n = 0 , on ∂ Ω   tricky case...see Babuˇ ska Paradox. 5 of 18

Polyharmonic operators For m ≥ 1, consider ( − 1 ) m ∆ m u = λ u in Ω 6 of 18

Polyharmonic operators For m ≥ 1, consider ( − 1 ) m ∆ m u = λ u in Ω Dirichlet boundary conditions for this problem  u = 0 , on ∂ Ω ,    ∂ k u ∂ n k = 0 , on ∂ Ω , 1 ≤ k ≤ m − 1    6 of 18

Weak formulation of the problem 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) We consider the eigenvalue problem in the weak form 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) We consider the eigenvalue problem in the weak form � � � A αβ D α uD β ϕ dx = λ u ϕ dx , ∀ ϕ ∈ V (Ω) Ω Ω | α | = | β | = m 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) We consider the eigenvalue problem in the weak form � � � A αβ D α uD β ϕ dx = λ u ϕ dx , ∀ ϕ ∈ V (Ω) Ω Ω | α | = | β | = m If V (Ω) = W m , 2 (Ω) 0 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) We consider the eigenvalue problem in the weak form � � � A αβ D α uD β ϕ dx = λ u ϕ dx , ∀ ϕ ∈ V (Ω) Ω Ω | α | = | β | = m If V (Ω) = W m , 2 (Ω) we talk about Dirichlet boundary conditions 0 7 of 18

Weak formulation of the problem The coefficients A αβ are fixed bounded real-valued functions defined on R N A αβ = A βα and � | α | = | β | = m A αβ ξ α ξ β ≥ θ | ξ | 2 V (Ω) is a closed subspace of W m , 2 (Ω) containing W m , 2 (Ω) , 0 compactly embedded into L 2 (Ω) We consider the eigenvalue problem in the weak form � � � A αβ D α uD β ϕ dx = λ u ϕ dx , ∀ ϕ ∈ V (Ω) Ω Ω | α | = | β | = m If V (Ω) = W m , 2 (Ω) we talk about Dirichlet boundary conditions 0 If V (Ω) = W m , 2 (Ω) 7 of 18