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if I have a sequence $f_k\in W_{1,p}(\Omega)$ which converge weakly to some function $ f $

and I know that $\nabla f_k-\nabla f\to 0$ in $L_{p}^{loc}(\Omega)$

I try to estimate the integral $\int_{\tilde{\Omega}} |\nabla f_k|^p-|\nabla f|^p$

$\tilde{\Omega}\subset \Omega$ (actually I try to prove it is 0 if $dist(\tilde{\Omega},\partial \Omega)>0 $

is there any suggestion for how to do this?

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    $\begingroup$ Since the restriction of gradient to $\widetilde {\Omega}$ converges in $L^p$, the $L^p$ norms converge. $\endgroup$ – user147263 Mar 11 '15 at 4:42
  • $\begingroup$ this is what I wanted to prove, what theorem you rely on? $\endgroup$ – Lin Mar 11 '15 at 11:33
  • $\begingroup$ Triangle inequality for the norm. $\endgroup$ – user147263 Mar 11 '15 at 16:11
  • $\begingroup$ the convergence is weak $\endgroup$ – Lin Mar 12 '15 at 7:49
  • $\begingroup$ But you have the strong convergence of the gradient on $\tilde\Omega$. $\endgroup$ – gerw Mar 13 '15 at 7:29

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