  We study properties of the classical fractional Sobolev spaces (or Bessel
potential spaces) on non-Lipschitz subsets of $\mathbb{R}^n$. We investigate
the extent to which the properties of these spaces, and the relations between
them, that hold in the well-studied case of a Lipschitz open set, generalise to
non-Lipschitz cases. Our motivation is to develop the functional analytic
framework in which to formulate and analyse integral equations on non-Lipschitz
sets. In particular we consider an application to boundary integral equations
for wave scattering by planar screens that are non-Lipschitz, including cases
where the screen is fractal or has fractal boundary.
