A question about Sobolev space and the relationship between the weak convergence on it and the strong convergence on L^p space.

Question

Suppose that $\Omega$ is a bounded open set in $\mathbb{R}^n$, $H_0^1(\Omega)$ is the Sobolev space $W_0^{1,2}(\Omega)$. If $u_n$ convergent to $u$ weakly in $H_0^1(\Omega)$, can I get the conclusion that $u_n$ convergent to $u$ strongly in $L^2(\Omega)$ and how to prove it? Or we can only get the conclusion with sense of a subsequence?

Thanks a lot for anyone and any help!

Answer 1

Suppose that $u_n$ does not converge to $u$ in $L^2(\Omega)$. Then, there exists $\varepsilon > 0$ and a subsequence such that $$ \|u_{n_k} - u\| \ge \varepsilon \qquad\forall k.$$ By compactness, there exists a further subsequence $u_{n_{k_\ell}}$ with $u_{n_{k_\ell}} \to v$ in $L^2(\Omega)$. By weak convergence in $H_0^1(\Omega)$, $u = v$. However, this contradicts the first inequality.