Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.Comment: This is a preprint version of a paper that will appear in the Proceedings of the 9th ISAAC Congress, Krak\'ow 201

Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi$ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.Comment: 34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math. Ser. (N.Y.), vol. 12, Springer, New York 201

Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain ω in ℝ N. We consider deformations Ø•(ω) of ω obtained by means of a locally Lipschitz homeomorphism Ø• and we estimate the variation of the eigenfunctions and eigenvalues upon variation of Ø•. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps Ø•. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains. © 2012 Copyright University College London

Monotonicity, continuity and differentiability results for the L pHardy constant

We consider the LpHardy inequality involving the distance to the boundary for a domain in the n-dimensional Euclidean space. We study the dependence on p of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such a constant is in general not explicitly known. © 2016, Hebrew University of Jerusalem

On an interior calder'on operator and a related steklov eigenproblem for maxwell's equations

We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calderón operator and an appropriate Dirichlet-to-Neumann map. The corresponding Neumann-to-Dirichlet map turns out to be compact, and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by G. Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calderón operator itself, and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity. © 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved