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Sobolev Spaces on Non-Lipschitz Subsets of \({\mathbb {R}}^n\) with Application to Boundary Integral Equations on Fractal Screens

Abstract

We study properties of the classical fractional Sobolev 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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Correspondence to D. P. Hewett.

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Chandler-Wilde, S.N., Hewett, D.P. & Moiola, A. Sobolev Spaces on Non-Lipschitz Subsets of \({\mathbb {R}}^n\) with Application to Boundary Integral Equations on Fractal Screens. Integr. Equ. Oper. Theory 87, 179–224 (2017). https://doi.org/10.1007/s00020-017-2342-5

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Keywords

  • Sobolev spaces
  • Non-Lipschitz sets
  • Integral equations