ISBN 9783843920971

978-3-8439-2097-1, Reihe Mathematik

Karsten Fritzsch
A Geometric Approach to Mapping Properties of Layer Potential Operators: The Cases of the Half-Space and of Two Touching Domains

161 Seiten, Dissertation Carl von Ossietzky Universität Oldenburg (2014), Softcover, B5

Zusammenfassung / Abstract

So far, no special framework for the study of layer potentials on manifolds with corners has been developed even though both approaches, the method of layer potentials and the calculus of conormal distributions on manifolds with corners, have been proven to be very successful. In this thesis, it is demonstrated in two important and representative special cases that the geometric viewpoint of singular geometric analysis leads to a feasible approach to the method of layer potentials: The Dirichlet and Neumann problems for Laplace's equation on the n-dimensional half-space are solved in spaces of polyhomogeneous functions and the more singular situation of two touching domains is studied. The latter correspond to a setting related to the phi-calculus and it is proven that the boundary layer potential operators are phi-pseudodifferential operators. Moreover, using the Push-Forward Theorem, we show that the relations between the layer potential operators and their boundary counterparts still hold in this setting and prove mapping properties of the layer potential operators between spaces of polyhomogeneous functions. We further improve these by using a local splitting of certain b-fibrations.