Chapter 2: Regularity Properties of Surface Potentials
DAVID L. COLTON, Rainer Kreß · Society for Industrial and Applied Mathematics eBooks · 2013
As the title of this book indicates, the first step in our analysis of the scattering of acoustic and electromagnetic waves by an obstacle is to reformulate the boundary-value problems of scattering theory as boundary integral equations. This will be accomplished by representing the solution of the boundary-value problem as a surface potential with respect to a given density, and then using the continuity properties of such potentials to arrive at the sought-after integral equation. Hence, in order to proceed with this objective, it is necessary to examine the regularity properties of surface potentials defined on closed surfaces. For the sake of brevity as well as practical importance, we shall restrict ourselves to surfaces in ℝ3 that are twice continuously differentiable. The extension of these results to Lyapunov surfaces in ℝn is straightforward (cf. Günter [1], Mikhlin [1]), although the problem of the scattering of waves by domains with corners presents new difficulties due to the loss of compactness of the associated integral operators (cf. Kleinman and Wendland [1], Wendland [3]).