Integral Equations and Operator Theory

, Volume 87, Issue 2, pp 179–224 | Cite as

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

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.

Keywords

Sobolev spaces Non-Lipschitz sets Integral equations 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  2. 2.Department of MathematicsUniversity College LondonLondonUK

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