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Mathematika

Table of Contents - July 2012 - Volume 58, Issue 02  

Research Article

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

Gerassimos Barbatisa1 and Pier Domenico Lambertia2

a1 Department of Mathematics, University of Athens, 157 84 Athens, Greece (email: gbarbatis@math.uoa.gr)

a2 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35121 Padova, Italy (email: lamberti@math.unipd.it)

Abstract

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain Ω in ℝ N . 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.

(Received May 12 2011)

(Online publication February 24 2012)

MSC (2010)

  • 35J25;
  • 47A75;
  • 47B25 (primary)