Eigenvalue problems in a non-Lipschitz domain

GABRIEL ACOSTA, MARÍA G. ARMENTANO

Londres

Congreso; MAFELAP 2013; 2013

Brunel University

In this work we analyze piecewise linear finite element approximations of the Laplace  eigenvalue  problem in the plane domain $Omega ={(x; y) : 0 < x < 1; 0 < y < x^{alpha}}$; which gives,   for $alpha >1 $, the simplest model of an external cusp. Since $Omega$ is curved and non- Lipschitz, our problem is not covered by the known literature which, as far as we know, only deals with polygonal or smooth domains. Indeed, the classical spectral theory can not be applied directly and in consequence we present the eigenvalue problem in a proper setting, and relying on known convergence results for the associated source problem with $1< alpha < 3$ (see cite{AA,AADL,AADL2}), we obtain quasi optimal order of convergence for the eigenpairs cite{AA2}.