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.