# Strong compactness in Sobolev spaces

Strong compactness in Sobolev spaces 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# Strong compactness in Sobolev spaces

, Volume 156 (4) – Sep 11, 2017
25 pages

/lp/springer_journal/strong-compactness-in-sobolev-spaces-jxQrtCgejb
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-017-0970-3
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Manuscripta MathematicaSpringer Journals

Published: Sep 11, 2017

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