Sobolev spaces are the standard framework in which to analyse weak (variational) formulations of PDEs or integral equations and their numerical solution (e.g. using the Finite Element Method or the Boundary Element Method). There are many different ways to define Sobolev spaces on a given domain, for example via integrability of weak derivatives, completions of spaces of smooth functions with respect to certain norms, or restriction from spaces defined on a larger domain. For smooth (e.g. Lipschitz) domains things many of these definitions coincide. But on rough (e.g. fractal) domains the picture is much more complicated. In this talk I'll try to give a flavour of the sort of interesting behaviour that can arise, and what implications this behaviour has for a "practical" example, namely acoustic wave scattering by fractal screens.
