The Helmholtz Equation on Lipschitz Domains

We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ( +k)u = 0 on a Lipschitz domain for all wave number k 2C with Imk 0. Following the approach for the case of smooth boundary [3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothness of a Lipschitz boundary brought some additional di culties. These are overcome through the use of harmonic analysis techniques together with a careful study of the properties of layer potentials near the boundary and the spectra of the traces of the layer potentials.