weak and strong convergence

Question

I don't know that why $||v_m||_{W^{1,p}_0}=1$, then we can assume that $v_m$ converges to $v_0$ weakly in ${W^{1,p}_0}(\Omega)$, and strongly in $L^p(\Omega)$

I can't understand clearly about weak and strong convergence, help me!

Answer 1

This under specific conditions is a consequence of Rellich-Kondrachov Theorem which states that for bounded $ \Omega \subset \mathbb{R}^n $ with Lipschitz boundary, the embedding $ W^{1,p}(\Omega) \rightarrow L^q(\Omega) $ is compact for $ 1 \leq q < p^* $ if $ 1 < p \leq n $, where $ 1/p^* = 1/p-1/n $. In other words if $ \|v_m\|_{W^{1,p}(\Omega)} < C $ then reducing to a weakly converging subsequence, we have $ v_0 \in W^{1,p}(\Omega) $ such that $ v_m \rightarrow v_0 $ in $ W^{1,p} $ and $ v_m \rightarrow v_0 $ in $ L^q $. Now if you have such conditions then you can easily observe that $ 1 < p < p^* $, so you can apply the theorem with $ q = p$.