Abstract:

This talk is concerned with the theory of boundary integral equations for Laplace's equation on Lipschitz domains. The theory for these equations in the space L^2(\Gamma), where \Gamma is the boundary of the domain, was developed in the 1980s by Calderon, Coifman, McIntosh, Meyer, and Verchota. However, the following question has remained open: can the standard second-kind integral equations, posed in L^2(\Gamma), be written as the sum of a coercive operator and a compact operator when \Gamma is only assumed to be Lipschitz, or even Lipschitz polyhedral? The practical importance of this question is that the convergence analysis the Galerkin method applied to these integral equations relies on this "coercive + compact" property holding. This talk will describe joint work with Simon Chandler-Wilde (University of Reading) that answers this question.