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In this paper, we define boundary single and double layer potentials for Laplace's equation in certain non-smooth bounded domains, more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Sobolev spaces and thus obtain layer potential solutions to the Dirichlet and Neumman problems with boundary data in these spaces. §1.. Introduction In this note we study layer potentials for Laplace's equation on the boundaries of certain bounded domains in Rn, n 3. As an application of our results, we obtain layer potential solutions to the Dirichlet and Neumann problems for the Laplacian. We remark that in Lipschitz domains and Lipschitz cylinders, there is an extensive literature concerning solution of the Dirichlet and Neumann problems for classical linear PDE arising in mathematical physics, e.g., Laplace's equation, the heat equation, Maxwell's equation, Stokes and Lame systems of equations (see (1), (2), (3), (5), (7), (10) and (12)). In this paper, we first consider a bounded domain satisfying the following condition: A1. There are positive constants d1,d2 and r0 such that for all 0 < r r0 and P 2 @ d1rn 1 H n 1( r(P)) d2rn 1,

Boundary integral operators and boundary value problems for Laplace’s equation