- 2009 (2) (entfernen)
- The outer oblique boundary problem of potential theory (2009)
- In this article 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 . Moreover we prove regularisation 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-Galerkinm approximation. The results are applicable to problems from Geomathematics, see e.g.  and .
- Limit Formulae and Jump Relations of Potential Theory in Sobolev Spaces (2009)
- In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from . Additionally, we need a result about the limit formulae in L2-norm, which can be found in , and a reduction result which we took from . Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. , from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also.