Explicit constants in Poincaré’s inequality, Sobolev extensions, and relations to boundary integral operators

Dr. Clemens Pechstein

Oct. 27, 2009, 3:30 p.m. P 215

Among the well-known constants in the theory of boundary integral equations are the ellipticity 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 domain, only little is known on their dependency on the shape of the boundary.

In this talk, we consider the homogeneous Laplace equation and derive explicit estimates for the above mentioned constants. It turns out that using an alternative trace norm, the dependency can be made explicit in two geometric paramters. One is the so-called Jones parameter. Jones proved extendability of Sobolev spaces for rather general domains, with a continuity constant explicit in this parameter. The other one is the constant in an isoperimetric inequality, which is equivalent to a Poincare type inequality. There are many domains with quite irregular, ragged boundaries, where both these parameters stay bounded.

There are applications to the analysis of both domain decomposition methods and BEM-based FEM.