Boundary Integral Equation Methods for Lipschitz Domains

Abstract

For the boundary value problems of Chaps. 3 and 4 we made assumptions which are often not met in applications. Indeed, the classical integral equation methods discussed in Chap. 3 require smoothness of the boundary ∂ D. In case of the cavity problem of Chap. 4 just a homogeneous boundary condition has been treated. Both restrictions are connected because if we like to weaken the regularity of the boundary, or if we like to allow for more general boundary conditions we have to investigate the traces of the functions or vector fields on the boundary ∂ D in detail. Therefore, we continue in Sects. 5.1.1 and 5.1.2 by introducing Sobolev spaces which appear as the range spaces of the trace operators and prove denseness, trace theorems and compact embedding results. Finally we use these results to extend the boundary integral equation methods for Lipschitz domains.