Implications of Weak convergence in Sobolev Spaces

Question

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in $L^{p}(\Omega;\mathbb{R}^{n})$? Or do we need stronger intial assumptions. Thanks.

Answer 1

Hint: Define $T:W^{1,p}(\Omega)\to L^p(\Omega)\times L^p(\Omega,\mathbb{R}^n)$ by $$Tu=(u,\nabla u)$$

With suitable norms on $W^{1,p}(\Omega)$ and $L^p(\Omega)\times L^p(\Omega,\mathbb{R}^n)$, you can prove that $T$ is an isometry.