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

Part of:General theory of linear operatorsSpecial classes of linear operatorsElliptic equations and systems

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)

Abstract

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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 equations47A75: Eigenvalue problems47B25: Symmetric and selfadjoint operators (unbounded)

Information

Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012, pp. 324 - 348
Copyright
Copyright © University College London 2012

References

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