Notes on Elliptic Boundary Value Problems for the Laplace Operator

In these notes we present the pseudodifferential approach to elliptic boundary value problems for the Laplace operator acting on functions on a smoothly bounded compact domain in a compact manifold. This is an elaboration of the classical method of multiple layer potentials. After a short discussion of this method we consider the theory of homogeneous distributions on R. This is useful in our subsequent discussion of boundary value problems and provides an interesting concrete complement to the rather abstract general theory developed earlier in the course. We then turn to boundary value problems. We analyze the smoothness and boundary regularity of multiple layer potentials for the Laplace equation. This allows the reduction of a boundary value problem to the solvability of a system of pseudodifferential equations on the boundary itself. After considering several different boundary value problems for smooth data we establish the Sobolev regularity properties of the single and double layer potentials. The estimates allow us to extend the existence results to data with finite differentiability and also establish the standard “elliptic estimates” for the solutions of elliptic boundary value problems for the Laplacian. This treatment is culled from material in L. Hörmander, The analysis of Linear Partial Differential Operators, III, M. Taylor, Partial Differential Equations, II and Introduction to the theory of Linear Partial Differential Equations by J. Chazarain and A. Piriou. I would finally like to thank Dara Cosgrove for the remarkable typing job.