Removable singularities for a Sobolev space

Abstract

Let Ω ⊂ R N 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 H 1 ( Ω ) ∩ C c ( Ω ¯ ) in H 1 ( Ω ) and H 1 ( 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.