We prove a strong compactness criterion in Sobolev spaces: given a sequence $(u_n)$ in $W_{\textrm{loc}}^{1,p}(\Rd)$, converging in $L_{\textrm{loc}}^{p}$ to a map $u\in W_{\textrm{loc}}^{1,p}(\Rd)$ and such that $|\n u_n | \le f$ almost everywhere, for some $f\in L_{\textrm{loc}}^{p}(\Rd)$, we provide a necessary and sufficient condition under which $(u_n)$ converges strongly to $u$ in $W_{\textrm{loc}}^{1,p}(\Rd)$. 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 $\n u_n \to \n u$ pointwise almost everywhere.

Strong compactness in Sobolev spaces / Zagatti, Sandro. - In: MANUSCRIPTA MATHEMATICA. - ISSN 0025-2611. - 156:3-4(2018), pp. 303-327. [10.1007/s00229-017-0970-3]

Strong compactness in Sobolev spaces

Zagatti, Sandro
2018-01-01

Abstract

We prove a strong compactness criterion in Sobolev spaces: given a sequence $(u_n)$ in $W_{\textrm{loc}}^{1,p}(\Rd)$, converging in $L_{\textrm{loc}}^{p}$ to a map $u\in W_{\textrm{loc}}^{1,p}(\Rd)$ and such that $|\n u_n | \le f$ almost everywhere, for some $f\in L_{\textrm{loc}}^{p}(\Rd)$, we provide a necessary and sufficient condition under which $(u_n)$ converges strongly to $u$ in $W_{\textrm{loc}}^{1,p}(\Rd)$. 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 $\n u_n \to \n u$ pointwise almost everywhere.
2018
156
3-4
303
327
https://link.springer.com/article/10.1007%2Fs00229-017-0970-3
Zagatti, Sandro
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/57131
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