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

Part of: Elliptic equations and systems Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  24 February 2012

Gerassimos Barbatis  and
Pier Domenico Lamberti
Gerassimos Barbatis
Affiliation:
Department of Mathematics, University of Athens, 157 84 Athens, Greece (email: gbarbatis@math.uoa.gr)
Pier Domenico Lamberti
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35121 Padova, Italy (email: lamberti@math.unipd.it)
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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.

MSC classification

Secondary: 35J25: Boundary value problems for second-order elliptic equations 47A75: Eigenvalue problems 47B25: Symmetric and selfadjoint operators (unbounded)
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 324 - 348
Copyright
Copyright © University College London 2012

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