Limit Formulae and Jump Relations of Potential Theory in Sobolev Spaces
- 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.