Removable singularities for a Sobolev space

AbstractLet 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 of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω 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