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

Published online by Cambridge University Press:  24 February 2012

Gerassimos Barbatis
Affiliation:
Department of Mathematics, University of Athens, 157 84 Athens, Greece (email: [email protected])
Pier Domenico Lamberti
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35121 Padova, Italy (email: [email protected])
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Abstract

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain Ω in ℝN. We consider deformations ϕ(Ω) of Ω obtained by means of a locally Lipschitz homeomorphism ϕ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of ϕ. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps ϕ. 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.

Type
Research Article
Copyright
Copyright © University College London 2012

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