Removable singularities for a Sobolev space |
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Authors: | Markus Biegert Mahamadi Warma |
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Institution: | Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany |
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Abstract: | Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H1(Ω) and H1(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide. |
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