London analysis and probability seminar

Abstract:

We study properties of fractional Sobolev spaces (Bessel potential spaces) on non-Lipschitz subsets of \R^n. We investigate the extent to which some of the standard properties of these spaces (e.g. density, duality and interpolation) that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. For instance, for a given smoothness parameter s\in\R, one might ask for which open sets \Omega\subset\R^n the set C^\infty_0(\Omega) is dense in the space of distributions in the Sobolev space H^s(\R^n) whose distributional support is contained in the closure of \Omega. This is always the case for Lipschitz (in fact, even for C^0) \Omega, but fails in general. One of our recent results is that the aforementioned density result holds for “thick” \Omega (in the sense of H. Triebel), which includes the Koch snowflake domain. Our studies are motivated by applications in PDE and integral equation theory, particularly to the analysis of boundary integral equation formulations of wave scattering by planar screens with fractal boundary. This is joint work with Simon Chandler-Wilde (Reading), Andrea Moiola (Pavia) and Antonio Caetano (Aveiro).