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Let $\Omega$ a bounded domain in $R^n$ with smooth boundary. I am reading a paper, and I have the following situation:

Consider $\varphi \in W^{1,p}(\Omega)$ and $v_j$ a sequence in $W^{1,p}(\Omega)$ such that $v_j - \varphi \in W^{1,p}_{0}(\Omega)$ for all $j$.

Suppose that $v_j - \varphi$ is a bounded sequence in $W^{1,p}_{0}(\Omega)$. The author says

Exist a function $u \in W^{1,p}(\Omega)$ with $ u - \varphi \in W^{1,p}_{0}(\Omega)$ and a subsequence of $v_j$ (that I willw denote by the sequence) such that

$$v_j \rightarrow u \ \text{in} \ W^{1,p}(\Omega) \ \textrm{weakly} \tag 1 $$

$$ v_j \rightarrow u \ \text{in} \ L^{p}(\Omega) \tag 2$$

$$ v_j \rightarrow u \ a.e \ \textrm{in} \ \Omega \tag 3$$

and the author says too : by the weak convergence

$$ \int_{\Omega} |\nabla u|^p \leq \liminf \int_{\Omega} |\nabla u_n|^p \tag 4$$

The parts (1), (2) and (3) I found a way of how to prove it, but I am not seeing how to prove part $(4)$. The best thing that I obtained is this (by the weak convergence) :

$$ \left( \int_{\Omega} |\nabla u|^p\right)^{1/p} +\left( \int_{\Omega} |u|^p\right)^{1/p} \leq \liminf \left[\int_{\Omega}\left( \int_{\Omega} |\nabla u_n|^p\right)^{1/p} +\left( \int_{\Omega} |u_n|^p\right)^{1/p} \right].$$

Someone could help me to justify the inequality (4)?

thank you

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  • $\begingroup$ what is $u_n$? can you define it? If you can show $u_n\to u$ weakly $W^{1,p}$ then inequality $(4)$ is the lower semi-continuity of sobolev space. $\endgroup$ – spatially Oct 26 '15 at 22:57
  • $\begingroup$ @tankonetoone: I think $u_n$ should be $v_j$. $\endgroup$ – gerw Oct 27 '15 at 14:32
  • $\begingroup$ @gerw I agree. Then your answer indeed makes sense $\endgroup$ – spatially Oct 27 '15 at 14:56
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This answer assumes that the $u_n$ in the question should be $v_j$.

The function $F : W^{1,p}(\Omega)$, $$f \mapsto \big( \int_\Omega | \nabla f|^p \, \mathrm{d}x\big)^{1/p}$$ is convex and continuous, hence, weakly lower semicontinuous. Your inequality (4) follows.

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