All Questions

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map $$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...

How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?

What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...