Removable singularities for a Sobolev space

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 H˜ 1(Ω) and H˜ 1(Ω \ F) coincide, that is, F is a removable singularity for H˜ 1(Ω). Here H˜ 1(Ω) is the closure of H1(Ω) ∩ Cc(Ω¯ ) in H1(Ω) and H1(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space H˜ 1(Ω) is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary ofΩ has finite (N −1)- dimensional Hausdorff measure, then our results show that we may replaceΩ by the better set Int(Ω¯ ) (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on Int(Ω¯ ) coincide.  2005 Elsevier Inc. All rights reserved.