Sobolev Spaces and Regularity for Polyhedral Domains

We prove a regularity result on polyhedral domains P ⊂ R using the weighted Sobolev spaces Ka (P). In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. In the proof, we identify Ka (P) with the Sobolev spaces on P associated to the metric r P gE , where gE is the Euclidean metric and rP(x) is a smoothing of the Euclidean distance from x to the set of singular points of P. A suitable compactification of the interior of P then becomes a compact manifold with corners with a distinguished class of vector fields (a Lie manifold). We then prove a tubular neighborhood theorem for Lie submanifolds. This allows us to extend most of the classical results on Sobolev spaces to weighted Sobolev spaces on polyhedral domains, including elliptic regularity. As an application, we include a well-posedness result for a non-standard boundary value problem on a smooth domain with boundary O using weighted Sobolev spaces, where the weight is the distance to the boundary.