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
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.