Shape-explicit constants for some boundary integral operators

Abstract

Among the well-known constants in the theory of boundary integral equations are the coercivity constants of the single-layer potential and the hypersingular boundary integral operator, and the contraction constant of the double-layer potential. Whereas there have been rigorous studies how these constants depend on the size and aspect ratio of the underlying domain, only little is known on their dependency on the shape of the boundary. In this article, we consider the homogeneous Laplace equation and derive explicit estimates for the above-mentioned constants in three dimensions. Using an alternative trace norm, we make the dependency explicit in two geometric parameters, the so-called Jones parameter and the constant in Poincaré's inequality. The latter one can be tracked back to the constant in an isoperimetric inequality. There are many domains with quite irregular boundaries, where these parameters stay bounded. Our results provide a new tool in the analysis of numerical methods for boundary integral equations and in particular for boundary element based domain decomposition methods.

Keywords:

• boundary integral equations
• boundary integral operators
• explicit constants
• Poincaré's inequality
• Sobolev extension
• boundary element method

AMS Subject Classifications::

• 65R20
• 65N38
• 31C15

Acknowledgements

The author would like to express his thanks to Olaf Steinbach (Graz University of Technology, Austria) and Paul Müller (Johannes Kepler University of Linz) for many fruitful discussions. Furthermore, the author is very grateful to Hyea Hyun Kim (Kyung Hee University, Korea) for providing and discussing the proof of Lemma 3.4 (Poincaré's inequality) in three dimensions. Last but not the least, the financial support by the Austrian Science Fund (FWF) under grants P19255-N18 and W1214 is gratefully acknowledged.