Limit formulae and jump relations of potential theory in Sobolev spaces

Abstract

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 on ∂Σ for integrable functions, see for example Fichera (Ann Math Pura ed Appl Serie IV, Tomo 27, 1948) or Freeden and Kersten (The geodetic boundary value problem using the known surface of the Earth. Veröffentlichung des Geodätischen Instituts der RWTH Aachen, 29, 1980). The achievement of this paper is the L ²(∂Σ) convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and ∂Σ is a two dimensional suitable smooth submanifold in ${\mathbb{R}^3}$, called regular C ^m,α-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 C ^m(∂Σ) which is proved with help of some results taken from Günter (Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik. Teubner, Leipzig, 1957). Additionally, we need a result about the limit formulae in L ²(∂Σ), which can be found in Kersten (Result Math 3:17–24, 1980), and a reduction result which we took from Müller (Math Ann 123:235–262, 1951). Moreover we prove the convergence in the Hölder spaces C ^m,β(∂Σ). Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize density results, see e.g. Freeden and Michel (Multiscale potential theory. Birkhäuser, Boston, 2004), from L ²(∂Σ) to H ^m,2(∂Σ). For it we prove the limit formula for U ₁ in (H ^m,2(∂Σ))^' also. The last section is dedicated to oblique limit formulae for the single layer potential as well as for its first order oblique derivative.