Weak convergence in Sobolev space $W^{1,p}$ implies strong convergence in $L^p$

Question

Let $p\neq \infty$ and let $\Omega\subseteq \mathbb{R}^N$ be a regular domain (a bounded open set with $C^1$ boundary). I want to prove that

If $\{u_n\}$ weakly converges in $W^{1,p}(\Omega)$ then it converges in $L^p(\Omega)$.

My attempt

By definition if $\{u_n\}$ weakly converges in $W^{1,p}(\Omega)$ then it weakly converges in $L^p(\Omega)$ too and this implies (by Banach-Steinhaus theorem) that $\{u_n\}$ is bounded in $L^p(\Omega)$. The same reasoning applies to the gradient of $u$ so $\{u_n\}$ is bounded in $W^{1,p}(\Omega)$.

By Rellich Kondrachov theorem there exists a subsequence of $\{u_n\}$ that converges in $L^p(\Omega)$. So I basically have

• $u_n\rightharpoonup u$ in $L^p(\Omega)$

• $u_{n_k}\to u$ in $L^p(\Omega)$

Is this enough to establish the strong convergence? If yes why?

Secondly, I was trying to construct a counterexample if $p=\infty$, but without any success. Could you help me?

Answer 1

The proof is false, since the Rellich Kondrachov theorem has two different exponents: $W^{1,p}\subset\subset L^q$ for all $1\leq q<p^*$, assuming $1\leq p<n$ ($p=\infty$ is not allowed! If you want a counterexample, look at the counterexample for the Rellich Kondrachov Theorem)

To complete the proof, one only needs to know that the inclusion map in the Rellich Kondrachov Theorem is compact: $W^{1,p}\subset\subset L^q$. Therefore, weakly $W^{1,p}$ convergent sequences are mapped to strongly $L^q$ convergent sequences. This is what you wanted to prove. If you want to understand this fact about compact operators, look at this proof.

Answer 2

There are actually two versions of the Rellich-Kondrachov theorem. The second one on that wiki page is just the answer to the question. The first version is the following: $W^{1,p}(\Omega)$ is compactly embedded into $L^q(\Omega)$ if $q<p^*$ and $p<n$. Since $p^*>p$ this answers the question for $p<n$.

For $p\ge n$, one can use that $W^{1,p}(\Omega)$ embeds (trivially) into $W^{1,n-\epsilon}(\Omega)$. Now the first version of Rellich-Kondrachov is applicable, and yields the compact embedding into $L^p(\Omega)$ provided $\epsilon$ is small enough. This uses that $p^* \to \infty$ for $p\nearrow n$.

This shows that the embedding operator of $W^{1,p}(\Omega)$ into $L^p(\Omega)$ is compact for all $p$, and so sequences converging weakly in $W^{1,p}(\Omega)$ converge strongly in $L^p(\Omega)$.