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EXISTENCE AND UNIQUENESS FOR A SEMILINEAR ELLIPTIC PROBLEM ON LIPSCHITZ DOMAINS IN RIEMANNIAN MANIFOLDS

ABSTRACT For a general class of semilinear equations in a Lipschitz domain , 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 functions. Results of this type for the linear equation are typically obtained using singular integral techniques.