A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator (original) (raw)
Journal of Differential Equations, 2005
Let Ω be an open connected subset of R n for which the Poincaré inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of R n , where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient φ(Ω) |Dv| 2 dy φ(Ω) |v| 2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W 1,2 0 (Ω) into the space L 2 (Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.
Zeitschrift Fur Analysis Und Ihre Anwendungen, 2005
Let Ω be an open connected subset of R n for which the Poincaré inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of R n , where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient φ(Ω) |Dv| 2 dy φ(Ω) |v| 2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W 1,2 0 (Ω) into the space L 2 (Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.
Spectral Properties of the Dirichlet-To-Neumann Operator on Lipschitz Domains
2007
The Dirichlet-to-Neumann operator D! is defined on L 2 (!) where ! is the boundary of a Lipschitz domain " and ! a real number which is not an eigenvalue of the Dirichlet Laplacian on L2("). We show that D! is a selfadjoint lower bounded operator with compact resolvent. There is a close connection between its eigenvalues and those of the Laplacian # µ on L 2 (") with Robin boundary conditions "u = µu|! where µ ! R. This connection is used to generalize L. Friedlander's result ! N+1 " ! D ,k =1 ,2 (where ! D is the k # th Dirichlet and ! N the k # th Neumann eigenvalue) to Lipschitz domains. We show that this Euclidean result is false, though, if an arbitrary compact Riemannian manifold M is considered instead of R d and " is suitable domain in M.
The spectral estimates for the Neumann–Laplace operator in space domains
Advances in Mathematics
In this paper we prove discreteness of the spectrum of the Neumann-Laplacian (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Sobolev-Poincaré inequalities that are obtained with the help of a geometric theory of composition operators on Sobolev spaces. These composition operators are induced by generalizations of conformal mappings that is called as mappings of bounded 2-distortion (weak 2-quasiconformal mappings).
A generic property for the eigenfunctions of the Laplacian
Topological Methods in Nonlinear Analysis, 2002
In this work we show that, generically in the set of C 2 bounded regions of R n , n ≥ 2, the inequality R Ω φ 3 = 0 holds for any eigenfunction of the Laplacian with either Dirichlet or Neumann boundary conditions.
Contemporary Mathematics, 2005
Let Ω ⊂ R d be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator -1 2 ∆ in Ω -Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ 0 (Ω, A) > 0 denote its principal eigenvalue. We discuss the question of monotonicity of λ 0 (Ω, A) in its dependence on the domain Ω. The main point of this note is to suggest an open problem that is in the spirit of Chavel's question concerning domain monotonicity for the Neumann heat kernal. Let Ω ⊂ R d be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator -1 2 ∆ in Ω -Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ 0 (Ω, A) > 0 denote its principal eigenvalue. If instead of the Neumann boundary condition, one imposes the Dirichlet boundary condition at ∂Ω, then it's easy to see that λ 0 (Ω, A) is monotone decreasing in Ω and increasing in A. Similarly, in the case at hand, it is clear that λ 0 (Ω, A) is monotone increasing in A; however, the question of monotonicity in Ω is not easily resolved. The impetus for studying this question arose in part from a recent paper in which one can find the asymptotic behavior of λ 0 (Ω, A) when A is a ball that shrinks to a point,