Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.