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?