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 onLpfor sharp ranges ofp's. Also, a detailed discussion of the Helmholtz eigenvalues of Lipschitz domains is presented.
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