On the spectral behavior of the Neumann Laplacian under mass density perturbation 9th ISAAC Congress Krakow, August 5 - 9, 2013 Luigi Provenzano, joint work with P. D. Lamberti Department of Mathematics. Doctoral School in Mathematical Sciences, Mathematics Area
Introduction Let Ω be a domain in R N of finite measure. Let ρ ∈ R := { f ∈ L ∞ (Ω) : ess inf Ω f ( x ) > 0 } ISAAC 2013 - 2 of 32
Introduction Let Ω be a domain in R N of finite measure. Let ρ ∈ R := { f ∈ L ∞ (Ω) : ess inf Ω f ( x ) > 0 } Consider � ( − 1 ) α D α � A αβ D β u � L u = 0 ≤| α | , | β |≤ m ISAAC 2013 - 2 of 32
Introduction Let Ω be a domain in R N of finite measure. Let ρ ∈ R := { f ∈ L ∞ (Ω) : ess inf Ω f ( x ) > 0 } Consider � ( − 1 ) α D α � A αβ D β u � L u = 0 ≤| α | , | β |≤ m and the eigenvalue problem L u = λρ u subject to homogeneous boundary conditions (Dirichlet, Neumann, intermediate, etc.) ISAAC 2013 - 2 of 32
Introduction � � � A αβ D α uD β ϕ dx = λ Q [ u , ϕ ] := u ϕρ dx ∀ ϕ ∈ V (Ω) Ω Ω 0 ≤| α | , | β |≤ m ISAAC 2013 - 3 of 32
Introduction � � � A αβ D α uD β ϕ dx = λ Q [ u , ϕ ] := u ϕρ dx ∀ ϕ ∈ V (Ω) Ω Ω 0 ≤| α | , | β |≤ m V (Ω) ⊂ H m (Ω) closed with V (Ω) ⊂ L 2 (Ω) compact; A αβ ∈ L ∞ (Ω) with A αβ = A βα ; there exist a , b , c > 0 such that a � u � 2 H m (Ω) ≤ Q [ u , u ] + b � u � 2 L 2 (Ω) , Q [ u , u ] ≤ c � u � 2 H m (Ω) ; ISAAC 2013 - 3 of 32
Poly-harmonic operators ( − ∆) m u = λρ u Let 0 ≤ k ≤ m and V (Ω) = H m (Ω) ∩ H k 0 (Ω) . ISAAC 2013 - 4 of 32
Poly-harmonic operators ( − ∆) m u = λρ u Let 0 ≤ k ≤ m and V (Ω) = H m (Ω) ∩ H k 0 (Ω) . k = m Dirichlet boundary conditions, V (Ω) = H m 0 (Ω) (N=2, m=2 clamped plate); 0 < k < m Intermediate boundary conditions, V (Ω) = H m (Ω) ∩ H k 0 (Ω) (N=2, m=2 hinged plate); k = 0 Neumann-type boundary conditions, V (Ω) = H m (Ω) (N=2, m=2 free vibrating plate). ISAAC 2013 - 4 of 32
The eigenvalue problem Our problem has a divergent sequence of eigenvalues − b < λ 1 [ ρ ] ≤ λ 2 [ ρ ] ≤ · · · ≤ λ j [ ρ ] ≤ · · · ISAAC 2013 - 5 of 32
The eigenvalue problem Our problem has a divergent sequence of eigenvalues − b < λ 1 [ ρ ] ≤ λ 2 [ ρ ] ≤ · · · ≤ λ j [ ρ ] ≤ · · · Our aim is to study the dependence ρ �→ λ j [ ρ ] ISAAC 2013 - 5 of 32
Analiticity of the eigenvalues Theorem Let F be a nonempty finite subset of N and let R [ F ] := { ρ ∈ R : λ j [ ρ ] � = λ l [ ρ ] , ∀ j ∈ F , l ∈ N \ F } , Θ[ F ] := { ρ ∈ R [ F ] : λ j 1 [ ρ ] = λ j 2 [ ρ ] , ∀ j 1 , j 2 ∈ F } . ISAAC 2013 - 6 of 32
Analiticity of the eigenvalues Theorem Let F be a nonempty finite subset of N and let R [ F ] := { ρ ∈ R : λ j [ ρ ] � = λ l [ ρ ] , ∀ j ∈ F , l ∈ N \ F } , Θ[ F ] := { ρ ∈ R [ F ] : λ j 1 [ ρ ] = λ j 2 [ ρ ] , ∀ j 1 , j 2 ∈ F } . Then R [ F ] is open in L ∞ (Ω) and the symmetric functions of the eigenvalues � Λ F , h [ ρ ] = λ j 1 [ ρ ] · · · λ j h [ ρ ] , h = 1 , . . . , | F | j 1 ,..., j h ∈ F j 1 < ··· < j h are analytic in R [ F ] . ISAAC 2013 - 6 of 32
Derivatives of the eigenvalues Theorem Let F be a nonempty finite subset of N . If F = ∪ n k = 1 F k and ρ ∈ ∩ n k = 1 Θ[ F k ] is such that for each k = 1 , ..., n the eigenvalues λ j [ ρ ] assume the common value λ F k [ ρ ] for all j ∈ F k , then the differential of Λ F , h at ρ is given by the formula n � ( u l ) 2 ˙ � � d Λ F , h [ ρ ][ ˙ ρ ] = − c k ρ dx , Ω k = 1 l ∈ F k ρ ∈ L ∞ (Ω) , where for each k = 1 , ..., n, { u l } l ∈ F k is an for all ˙ orthonormal basis in L 2 ρ (Ω) of the eigenspace associated with λ F k [ ρ ] . ISAAC 2013 - 7 of 32
Critical mass densities We assume V (Ω) ⊂ H 1 0 (Ω) ISAAC 2013 - 8 of 32
Critical mass densities We assume V (Ω) ⊂ H 1 0 (Ω) � � � Let M > 0 and L M := ρ ∈ R : Ω ρ dx = M . ISAAC 2013 - 8 of 32
Critical mass densities We assume V (Ω) ⊂ H 1 0 (Ω) � � � Let M > 0 and L M := ρ ∈ R : Ω ρ dx = M . Theorem Let F be a nonempty finite subset of N . Then for all h = 1 , ..., | F | the function which takes ρ ∈ R [ F ] ∩ L M to Λ F , h [ ρ ] has no critical mass densities ˜ ρ such that λ j [˜ ρ ] � = 0 and have the same sign for all j ∈ F. ISAAC 2013 - 8 of 32
Critical mass densities We assume V (Ω) ⊂ H 1 0 (Ω) � � � Let M > 0 and L M := ρ ∈ R : Ω ρ dx = M . Theorem Let F be a nonempty finite subset of N . Then for all h = 1 , ..., | F | the function which takes ρ ∈ R [ F ] ∩ L M to Λ F , h [ ρ ] has no critical mass densities ˜ ρ such that λ j [˜ ρ ] � = 0 and have the same sign for all j ∈ F. n � � u 2 c k l = const = ⇒ u 1 = ... = u | F | = 0 k = 1 l ∈ F k ISAAC 2013 - 8 of 32
Maximum principle Theorem Let C ⊂ L ∞ (Ω) be a bounded set. Then the functions from C to R which take ρ ∈ C to λ j [ ρ ] are weakly* continuous for all j ∈ N . ISAAC 2013 - 9 of 32
Maximum principle Theorem Let C ⊂ L ∞ (Ω) be a bounded set. Then the functions from C to R which take ρ ∈ C to λ j [ ρ ] are weakly* continuous for all j ∈ N . Theorem Let C ⊆ R [ F ] be a weakly* compact subset of L ∞ (Ω) . Let M > 0 such that C ∩ L M is not empty. Assume that all the eigenvalues λ j [ ρ ] have the same sign and do not vanish for all j ∈ N , ρ ∈ C. Then for all h = 1 , ..., | F | the function which takes ρ ∈ C ∩ L M to Λ F , h [ ρ ] has maxima and minima, and such points belong to ∂ C ∩ L M . ISAAC 2013 - 9 of 32
Neumann boundary conditions Let Ω be a bounded domain in R N of class C 1 . The eigenvalue problem for the Laplacian with Neumann boundary conditions is � − ∆ u = λρ u , in Ω , (1) ∂ u ∂ν = 0 , on ∂ Ω . ISAAC 2013 - 10 of 32
Neumann boundary conditions Let Ω be a bounded domain in R N of class C 1 . The eigenvalue problem for the Laplacian with Neumann boundary conditions is � − ∆ u = λρ u , in Ω , (1) ∂ u ∂ν = 0 , on ∂ Ω . We have a sequence 0 < λ 1 [ ρ ] ≤ λ 2 [ ρ ] ≤ · · · ≤ λ j [ ρ ] ≤ · · · ISAAC 2013 - 10 of 32
Critical mass densities Theorem Let Ω be a bounded domain in R N of class C 1 , F = { m , n } , with m , n ∈ N , m � = n. Let ˜ ρ ∈ R [ F ] continuous, such that the solutions of (1) be classic solutions and moreover their nodal domains are stokians. Then for h = 1 , 2 , ˜ ρ is not a critical mass density for the function which takes ρ ∈ R [ F ] ∩ L M to Λ F , h [ ρ ] . Moreover all simple eigenvalues have no critical mass densities under the fixed mass constraint. � c i u 2 i = const i ∈ F ISAAC 2013 - 11 of 32
Critical mass densities Theorem Let Ω ⊂ R N and F be as in Theorem 6. Let C ⊆ R [ F ] be a weakly* compact subset of L ∞ (Ω) . Let M > 0 and L M = { ρ ∈ L ∞ (Ω) : � Ω ρ = M } . Then for h = 1 , 2 , the function which takes ρ ∈ C ∩ L M to Λ F , h [ ρ ] admits points of maximum and minimum, and if for such points the solutions of problem (1) are classic solution, they belong to ∂ C ∩ L M . ISAAC 2013 - 12 of 32
Steklov boundary conditions Let Ω be a bounded domain in R N of class C 1 . The eigenvalue problem for the laplacian with Steklov boundary condition is � ∆ u = 0 , in Ω , (2) ∂ u ∂ν = λρ u , on ∂ Ω . ρ ∈ R ′ := { f ∈ L ∞ ( ∂ Ω) : ess inf ∂ Ω f ( x ) > 0 } . ISAAC 2013 - 13 of 32
Steklov boundary conditions Let Ω be a bounded domain in R N of class C 1 . The eigenvalue problem for the laplacian with Steklov boundary condition is � ∆ u = 0 , in Ω , (2) ∂ u ∂ν = λρ u , on ∂ Ω . ρ ∈ R ′ := { f ∈ L ∞ ( ∂ Ω) : ess inf ∂ Ω f ( x ) > 0 } . We have a sequence 0 < λ 1 [ ρ ] ≤ λ 2 [ ρ ] ≤ · · · ≤ λ j [ ρ ] ≤ · · · ISAAC 2013 - 13 of 32
Analyticity of eigenvalues and derivatives Theorem Let Ω be a bounded domain in R N of class C 1 and F a nonempty finite subset of N . Then the symmetric functions of eigenvalues Λ F , h are analytic in R [ F ] . Moreover, if ρ ∈ Θ[ F ] and the eigenvalues λ j [ ρ ] assume the common value λ F [ ρ ] for all j ∈ F, then the differential of Λ F , h at ρ is given by the formula � | F | − 1 � � � ρ ] = − ( λ F [ ρ ]) h + 1 u 2 d Λ F , h [ ρ ][ ˙ l ˙ ρ d σ , h − 1 ∂ Ω l ∈ F ρ ∈ L ∞ ( ∂ Ω) , where { u l } is a hortonormal basis for λ F [ ρ ] in for all ˙ H 1 , 0 � u ∈ H 1 (Ω) : � � ρ (Ω) := ∂ Ω u ρ d σ = 0 . ISAAC 2013 - 14 of 32
Critical mass densities Proposition Let B = B N ( 0 , 1 ) be the unit ball in R N , S N the ( N − 1 ) -dimensional measure of ∂ B, F = { 1 , ..., N } , M > 0 . Then the constant mass density ρ M = M S N is a critical mass density for � Λ F , h for h = 1 , ..., N under the constraint ∂ Ω ρσ = M. ISAAC 2013 - 15 of 32
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