The Analysis of Oblique Boundary Problems and Limit Formulae Motivated by Problems from Geomathematics ; Die Analysis der schiefachsiger Randwertprobleme und der Grenzwertgleichungen motiviert durch geomathematische Probleme
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Media type:
Doctoral Thesis;
Electronic Thesis;
E-Book
Title:
The Analysis of Oblique Boundary Problems and Limit Formulae Motivated by Problems from Geomathematics ; Die Analysis der schiefachsiger Randwertprobleme und der Grenzwertgleichungen motiviert durch geomathematische Probleme
Contributor:
Raskop, Thomas
[Author]
Published:
KLUEDO - Publication Server of University of Kaiserslautern-Landau (RPTU), 2009
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic ...