Bounded sequences in $H_0^1(0,1)$

Question

Let $\{u_n\}_n$ be a bounded sequence on $H_0^1(0,1)$. Can we find a subsequence such that $u_{n_k}'(x)\longrightarrow u'(x)\text{ a.e. }x\in(0,1)$ for some $u\in H_0^1(0,1)$?

Answer 1

Here's a suggestion for a counterexample:

For the sake of simplicity, I changed the interval to $(0, \pi)$, otherwise you can shift everything. Define:

$$ u_n (x) := \frac{\sin(nx)}{n} $$

Then $ (u_n)_n$ is bounded in $ H^{1}_0(0, \pi)$ and $ u'_n(x)= \cos(nx) $. But such sequence does not admit any a.e convergent subsequence, as you can see here (discussion made for sine, but the idea is the same); in particular, there is no $u \in H^1_0(0, \pi)$ such that $u'_{n_k} \rightarrow u'$ almost everywhere.