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.
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