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

Integral Equations and Operator Theory volume 87, pages 179–224 (2017)Cite this article

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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    S. N. Chandler-Wilde & A. Moiola

  2. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    D. P. Hewett

Authors
  1. S. N. Chandler-Wilde
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  2. D. P. Hewett
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  3. A. Moiola
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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