On the oblique boundary problem with a stochastic inhomogeneity

Abstract

We analyze the regular oblique boundary problem for the Poisson equation on a C 1,1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem, we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax–Milgram lemma enables us to prove an existence and uniqueness result for the weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then, we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case, we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples, we show the applicability of our results. Finally a Ritz–Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.