Stability of Sobolev spaces with zero boundary values (Q2712736)

Let \(\Omega\) be a bounded domain in \(\mathbb R^n\). Let \(1<p<\infty\) and let \(W_0^{1,p}(\Omega)\) be the closure of \(C_0^{\infty}(\Omega)\) in the \(W^{1,p}\)-norm defined by NEWLINE\[NEWLINE \|u\|_{1,p}^p= \int _{\Omega}(|u|^p +|\nabla u|^p) dx. NEWLINE\]NEWLINE In connection with the \(p\)-Laplace equation it is of interest to know for which bounded open domains \(\Omega\) it is true that NEWLINE\[NEWLINE W_0^{1,p}(\Omega) =W^{1,p}(\Omega)\cap \bigcap\limits_{q<p}W_0^{1,q}(\Omega). NEWLINE\]NEWLINE The paper under review contains a survey of the results obtained in \textit{L.~I.~Hedberg} and \textit{T.~Kilpeläinen}, Math. Scand. 85, 245-258 (1999; Zbl 0962.46018), results which solve the above mentioned problem in terms of capacities.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].