Oblique Boundary Value Problems and Limit Formulae of Potential Theory: an Analysis Motivated by Geomathematical Problems

This book deals with two main subjects which are of interest in the field of geomathematics. Here we search for harmonic functions satisfying certain boundary conditions. Often this is a condition on the derivative in direction of the gravity vector. Thus an oblique boundary problem occurs, because the gravity vector is not the normal vector of the earths surface. In this book we provide weak solutions to oblique boundary problems for very general coefficients, surfaces and inhomogeneities. We start with the problem for bounded domains solving a weak formulation for inhomogeneities in Sobolev spaces. Going to unbounded domains, we use the Kelvin transformation to get a corresponding bounded problem. Moreover Stochastic inhomogeneities, a Ritz Galerkin approximation and some examples are provided. The limit formulae are strongly related to boundary problems of potential theory, because they can be used to construct harmonic functions. We prove the formulae in several norms. The achievement is the convergence in Sobolev norms, which we use in order to prove the density of several geomathematical function systems in Sobolev spaces, e.g. the spherical harmonics.