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I have to solve the Exercise $10.43$ of page $302$ of the book "A First Course in Sobolev Space" by Leoni. The problem is the following.

Let $\Omega\subset\mathbb{R}^n$ be an open set, $1\leq p<+\infty$ and $\{u_n\}\subset W^{1, p}(\Omega)$. Prove that $u_n\rightharpoonup u$ in $W^{1, p}(\Omega)$ if and only if $u_n\rightharpoonup u$ in $L^p(\Omega)$ and $\nabla u_n\rightharpoonup\nabla u$ in $L^p(\Omega; \mathbb{R}^n)$.

Some ideas/hints?

Thank You

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Note that $W^{1,p}(\Omega)\to L^p(\Omega)\times L^p(\Omega;\mathbb R^n):u\mapsto(u,\nabla u)$ is an isometry (perhaps after changing to an equivalent norm). Can you see what to do now?

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  • $\begingroup$ How can I prove that the map $u\mapsto(u, \nabla u)$ is an isometry? Thank you @Jason $\endgroup$ – Redeldio Aug 12 '17 at 14:05
  • $\begingroup$ What norm do you have on $W^{1,p}$? If for example you are using $\|u\|_{W^{1,p}}:=\|u\|_p+\sum_{j=1}^n\|\partial_ju\|_p$, then give $L^p(\Omega)\times L^p(\Omega;\mathbb R^n)$ the norm $\|(u,v)\|:=\|u\|_p+\sum_{j=1}^n\|v_j\|_p$. Now it's automatic. $\endgroup$ – Jason Aug 13 '17 at 17:50
  • $\begingroup$ Ok, now I understand. Thank you $\endgroup$ – Redeldio Aug 14 '17 at 13:24

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