Boundary value problems and Hardy spaces associated to the Helmholtz equation in Lipschitz domains

Abstract

In this paper we introduce and discuss, in the Clifford algebra framework, certain Hardy-like spaces which are well suited for the study of the Helmholtz equation Δu + k2u = 0 in Lipschitz domains of ℝn + 1. In particular, in the second part of the paper, these results are used in connection with the classical boundary value problems for the Helmholtz equation in Lipschitz domains in arbitrary space dimensions. In this setting, existence, uniqueness, and optimal estimates are obtained by inverting the corresponding layer potential operators on Lp for sharp ranges of p's. Also, a detailed discussion of the Helmholtz eigenvalues of Lipschitz domains is presented. © 1996 Academic Press, Inc.