Characterization for the Kuratowski limits of a sequence of Sobolev spaces
Let \(B\) be the unit ball in \(\mathbb{R}^n\), \(\{\Omega_j\}^\infty_{j=1}\subset B\) and \(\Omega\subset B\) be domains. The author studies the Kuratowski-Mosco limits \[ \{u\in\mathring W^1_p(B): \exists u_j\in \mathring W^1_p(\Omega_j),u_j\to u\}, \] \[ \{u\in\mathring W^1_p(B): \exists u_{j_k}\in \mathring W^1_p(\Omega_{j_k}), u_{j_k}\to u\}, \] where \(\mathring W^1_p\) are the usual Sobolev spaces, \(1< p<\infty\). The convergence is in \(\mathring W^1_p(B)\). Criteria are formulated in terms of capacity. Applications to the continuity behaviour of solutions of the Dirichlet problem for the \(p\)-Laplacian are given.
scientific article; zbMATH DE number 1271362