Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Question

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ u_{k_j}\}$ converging to some $u$ in $L^p(U)$. My question is this: does $u$ necessarily have a (weak) derivative? I assume the answer is no, otherwise the theorem would probably include this. However, I am having trouble thinking of a counterexample. Can someone give an example of a bounded sequence in $W^{1,p}(U)$ which converges to a function in $L^p(U)$ which has no weak derivative?

Answer 1

Yes, it does. Since $W^{1,p}(U)$ is reflexive, there exists a further subsequence $\{u_{k_{j_i}}\}$ which converges weakly to some $v\in W^{1,p}(U)$. It must hence also converge weakly to $v$ in $L^p$, and since $u_{k_{j_i}}\to u$ in $L^p$ it follows that $u=v$ and hence has a weak derivative.