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arXiv · 1607.01994

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

Simon N. Chandler-Wilde·David P. Hewett·Andrea Moiola
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Abstract

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.

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Simon N. Chandler-Wilde, David P. Hewett, Andrea Moiola. 2016-07-07. Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens. https://doi.org/10.1007/s00020-017-2342-5

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