Strong compactness in Sobolev spaces

Abstract

We prove a strong compactness criterion in Sobolev spaces: given a sequence $(u_n)$ in $W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$, converging in $L_{\text {loc}}^{p}$ to a map $u\in W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$ and such that $|\nabla u_n | \le f$ almost everywhere, for some $f\in L_{\text {loc}}^{p}({\mathbb {R}}^d)$, we provide a necessary and sufficient condition under which $(u_n)$ converges strongly to u in $W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$. In addition we prove a pointwise version of the criterion, according to which, given $(u_n)$ and u as above, but with no boundedness assumptions on the sequence of gradients, we have $\nabla u_n \rightarrow \nabla u$ pointwise almost everywhere.