Integral Equations and Operator Theory

, Volume 87, Issue 2, pp 179–224

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

Article

DOI: 10.1007/s00020-017-2342-5

Cite this article as:
Chandler-Wilde, S.N., Hewett, D.P. & Moiola, A. Integr. Equ. Oper. Theory (2017) 87: 179. doi:10.1007/s00020-017-2342-5

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 

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