We prove a strong compactness criterion in Sobolev spaces: given a sequence $$(u_n)$$ ( u n ) in $$W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$$ W loc 1 , p ( R d ) , converging in $$L_{\text {loc}}^{p}$$ L loc p to a map $$u\in W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$$ u ∈ W loc 1 , p ( R d ) and such that $$|\nabla u_n | \le f$$ | ∇ u n | ≤ f almost everywhere, for some $$f\in L_{\text {loc}}^{p}({\mathbb {R}}^d)$$ f ∈ L loc p ( R d ) , we provide a necessary and sufficient condition under which $$(u_n)$$ ( u n ) converges strongly to u in $$W_{\text {loc}}^{1,p}({\mathbb {R}}^d)$$ W loc 1 , p ( R d ) . In addition we prove a pointwise version of the criterion, according to which, given $$(u_n)$$ ( u n ) and u as above, but with no boundedness assumptions on the sequence of gradients, we have $$\nabla u_n \rightarrow \nabla u$$ ∇ u n → ∇ u pointwise almost everywhere.
Manuscripta Mathematica – Springer Journals
Published: Sep 11, 2017
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