"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ $|u_j|^{p^*}\rightharpoonup d\nu$ weakly* in the sense of measures."
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2$\begingroup$ Please take the few extra moments to retype a short quote like this, instead of including a scanned image; it is much better for searching. Also please include a reference to the source. $\endgroup$– Nate EldredgeCommented Apr 27, 2015 at 20:57
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$\begingroup$ Also please keep your titles as descriptive as possible. "Surching a proof" conveys absolutely no information as to the topic of your question. I rolled back your edit. $\endgroup$– Nate EldredgeCommented Apr 27, 2015 at 21:27
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Assuming $1 < p < \infty$, $W^{1,p}(\Omega)$ is reflexive, so bounded sets are weakly precompact by Alaoglu's theorem (the weak-* and weak topologies coincide). Thus $u_j$ has a subsequence converging weakly to some $u \in W^{1,p}(\Omega)$. Now $W^{1,p}_0(\Omega)$ is convex and strongly closed in $W^{1,p}(\Omega)$, hence also weakly closed, and therefore $u \in W^{1,p}_0(\Omega)$.
If $u_j$ is bounded in $W^{1,p}$ then by definition $|\nabla u_j|^p$ is bounded in $L^1$. So the measures $|\nabla u_j|^p \,dm$ are bounded in total variation, i.e. norm-bounded as elements of $C_0(\Omega)^*$, which by Riesz representation is the space of Radon measures on $\Omega$. By Alaoglu again, the sequence is weak-* compact and so a subsequence converges weak-* to a measure $\mu$.
Likewise, by Sobolev embedding, $u_j$ is bounded in $L^{p^*}(\Omega)$ so $|u_j|^{p^*}$ is bounded in $L^1(\Omega)$ and a similar argument applies.
answered Apr 27, 2015 at 21:09
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$\begingroup$ @Vrouvrou: This is such a standard argument that I do not think you will find a reference that discusses it specifically. The results I mentioned will be in any elementary textbook on functional analysis and Sobolev spaces, such as Folland's Real Analysis for example. $\endgroup$ Commented Apr 27, 2015 at 21:19
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$\begingroup$ can i replace $L^{p^*}$ by an other space such that $W^{1,p}$ is continuously embeded to it but not compactly ? $\endgroup$– VrouvrouCommented Apr 27, 2015 at 21:29
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$\begingroup$ @Vrouvrou: Yes, the compact embedding is not used here. But your other space will also need to be continuously embedded in $C_0(\Omega)^*$. $\endgroup$ Commented Apr 27, 2015 at 21:30
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$\begingroup$ what is the space $C_0(\Omega)^*$ ? thank you $\endgroup$– VrouvrouCommented Apr 27, 2015 at 21:32
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$\begingroup$ @Vrouvrou: $C_0(\Omega)$ is the space of continuous functions on $\Omega$ which vanish at "infinity", i.e. the uniform closure of the continuous compactly supported functions $C_c(\Omega)$. For a bounded open set $\Omega$, this means continuous functions on $\bar{\Omega}$ that vanish on $\partial \Omega$. $C_0(\Omega)^*$ is the dual of this space, which as I said, by the Riesz representation theorem, is isometrically isomorphic to the space of signed Radon measures on $\Omega$, with the total variation norm. $\endgroup$ Commented Apr 27, 2015 at 21:35