Some eigenvalue problems in non-Lipschitz domains

GABRIEL ACOSTA, MARÍA G. ARMENTANO

Puerto Varas

Congreso; LANUS-FEM: On the occasion of Rodolfo Rodríguez?s 60th Birthday; 2014

The goal of this talk is the analysis of piecewise linear finite element approximations of  spectral problems in a plane domain $Omega ={(x; y) : 0 < x < 1; 0 < y < x^{alpha}}$;   which gives, for $alpha >1 $, the simplest model of an external cusp. This kind  of   problems has not been covered by the known literature which, as far as we know, only has dealt with polygonal or smooth domains. Indeed, since $Omega$ is curved and non-Lipschitz, the classical spectral theory can not be applied directly, in particular, certain classical results in Sobolev spaces do not hold in domains with external cusps. Nevertheless, for the spectral approximations of the Laplace operator with mixed boundary conditions, we can present the eigenvalue problem in a proper setting and relying on known convergence results for the associated source problem (see cite{AA,AADL,AADL2}) we can obtain quasi optimal order of convergence for the eigenpairs cite{AA2}.