Spectral boundary value problems for the Helmholtz equation with spectral parameter in boundary conditions on a non-smooth surface

Abstract

The spectral properties of four problems for the Helmholtz equation with spectral parameter in boundary or transmission conditions on a closed Lipschitz surface S are studied. These problems are related to the classical integral operators of potential type on S for the Helmholtz equation. They have been studied before in the case when S is infinitely smooth. It is shown that the most important properties of eigenvalues and root functions hold also for Lipschitz surfaces S. The machinery of potential theory in Lipschitz domains and of spectral theory is used in the proofs.