# The essential numerical range of the Laplace double-layer potential on Lipschitz domains

**Date**: Wednesday 27 March 2019, 16:30 – 17:30**Location**: Roger Stevens LT 14 (10M.14)**Type**: Analysis and Applications, Seminars, Pure Mathematics**Cost**: Free

#### Dr Euan Spence, University of Bath. Part of the Analysis and Applications Seminar Series.

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, e.g., 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 plus a compact operator when $\Gamma$ is only assumed to be Lipschitz, or even Lipschitz polyhedral?

These second-kind equations involve the double-layer potential and this “coercive + compact” property can be rephrased as the property that the essential numerical range of the double-layer potential does not contain plus or minus one. The practical importance of this question is that the convergence analysis of Galerkin discretisations of these integral equations relies on this “coercive + compact” property holding.

This talk will describe joint work with Professor Simon Chandler-Wilde (University of Reading) that answers this question.