Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds

For a general class of semilinear equations Deltau - F(x, u) = 0 in a Lipschitz domain Omega subset of M, where M is a smooth compact Riemannian manifold, we establish the existence and uniqueness of solutions to Dirichlet and Neumann boundary problems. The main contribution of this paper is that we consider 'rough' boundary data that are typically just L-p(partial derivativeOmega) functions. Results of this type for the linear equation are typically obtained using singular integral techniques.