Oblique Stochastic Boundary-Value Problem

Abstract

The aim of this chapter is to report the current state of the analysis for weak solutions to oblique boundary problems for the Poisson equation. In this chapter, deterministic as well as stochastic inhomogeneities are treated and existence and uniqueness results for corresponding weak solutions are presented. We consider the problem for inner bounded and outer unbounded domains in $\mathbb{R}^{n}$. The main tools for the deterministic inner problem are a Poincaré inequality and some analysis for Sobolev spaces on submanifolds, in order to use the Lax-Milgram lemma. The Kelvin transformation enables us to translate the outer problem to a corresponding inner problem. Thus, we can define a solution operator by using the solution operator of the inner problem. The extension to stochastic inhomogeneities is done with the help of tensor product spaces of a probability space with the Sobolev spaces from the deterministic problems. We can prove a regularization result, which shows that the weak solution fulfills the classical formulation for smooth data. A Ritz-Galerkin approximation method for numerical computations is available. Finally, we show that the results are applicable to geomathematical problems.