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We study properties of the classical fractional Sobolev spaces (or Besselpotential spaces) on non-Lipschitz subsets of $\mathbb{R}^n$. We investigatethe extent to which the properties of these spaces, and the relations betweenthem, that hold in the well-studied case of a Lipschitz open set, generalise tonon-Lipschitz cases. Our motivation is to develop the functional analyticframework in which to formulate and analyse integral equations on non-Lipschitzsets. In particular we consider an application to boundary integral equationsfor wave scattering by planar screens that are non-Lipschitz, including caseswhere the screen is fractal or has fractal boundary.

Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens