We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega )$ of $\Omega $ obtained by means of a locally Lipschitz homeomorphism $\phi $ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\phi $. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\phi$. 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.

SPECTRAL STABILITY ESTIMATES FOR ELLIPTIC OPERATORS SUBJECT TO DOMAIN TRANSFORMATIONS WITH NON-UNIFORMLY BOUNDED GRADIENTS

LAMBERTI, PIER DOMENICO
2012

Abstract

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega )$ of $\Omega $ obtained by means of a locally Lipschitz homeomorphism $\phi $ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\phi $. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\phi$. 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
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/2521119
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