The Outer Oblique Boundary Problem of Potential Theory

Abstract

In this article, we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients, and inhomogeneities. The main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in Grothaus and Raskop (Stochastics 2006; 78(4):233–257). Moreover we prove regularization results for the weak solutions of both the inner and the outer problem. We investigate the nonadmissible direction for the oblique vector field, state results with stochastic inhomogeneities, and provide a Ritz–Galerkin approximation. The results are applicable to problems from Geomathematics (see, e.g., Freeden and Maier (Electronic Trans. Numer. Anal. 2002; 14:40–62 and Bauer, An Alternative Approach to the Oblique Derivative Problem in Potential Theory, Shaker Verlag, Aachen, 2004)).

Keywords:

• Kelvin transformation
• Oblique boundary problem
• Poisson equation
• Sobolev spaces
• Submanifolds
• Weak solutions

Mathematics Subject Classification 2000:

• 31B20
• 35J05
• 35J25
• 46E35
• 32C25
• 86A30

ACKNOWLEDGMENTS

We thank Willi Freeden for inspiring discussions. Furthermore, financial support by the ISGS Kaiserslautern in the form of a fellowship of the German state Rhineland–Palatinate as well as financial support by the Mathematics Department of the University Kaiserslautern is gratefully acknowledged.