Existence of a subsequence that converges to a function weakly in $W^{2,p}$ and strongly in $L^p$

Question

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ with $C^1$ boundary and consider the Sobolev space $W^{k,p}(U)$ consisting of all locally summable functions $u:U\to\mathbb{R}$ such that for each multi-index $\alpha$ with $|\alpha|\leq k$, $D^\alpha u$ exists in the weak sense and belongs to the Lebesgue space $L^p(U)$. If $u\in W^{k,p}(U)$, we define its norm to be $$\Vert u\Vert_{W^{k,p}\ (U)}=\begin{cases} \left(\sum_{|\alpha|\leq k}\int_U|D^\alpha u|^p\right)^\frac{1}{p}&,1\leq p<\infty,\\ \sum_{|\alpha|\leq k}\mathrm{ess\ sup}_U|D^\alpha u|&,p=\infty. \end{cases}$$

Question: Can we extract a subsequence $\{u_{i_j}\}_{j=1}^\infty$ from a bounded sequence $\{u_i\}_{i=1}^\infty$ in $W^{2,p}(U)$ so that $\{u_{i_j}\}_{j=1}^\infty$ converges to $u$ weakly in $W^{2,p}(U)$ and strongly in $L^p(U)$?

Attempt: Since a bounded sequence in $W^{2,p}(U)$ is surely bounded in $W^{1,p}(U)$, the Rellich-Kondrachov theorem assures us of a subsequence $\{u_{i_j}\}_{j=1}^\infty$ that converges in $L^p(U)$ to some $u$. This is probably how the strong convergence in $L^p(U)$ is done, but I don't know what to do with $\{u_{i_j}\}_{j=1}^\infty$ to get the weak convergence in $W^{2,p}(U)$. How do these two convergences share the same limit $u$?

Would it be better to first find a weakly convergent subsequence $\{u_{i_j}\}_{j=1}^\infty$ in $W^{2,p}(U)$ and then extract a strongly convergent sequence $\{u_{i_{j_m}}\}_{m=1}^\infty$ in $L^p(U)$? More precisely, since $W^{2,p}(U)$ is reflexive, we can extract a subsequence $\{u_{i_j}\}_{j=1}^\infty$ that converges to some $u$ weakly in $W^{2,p}(U)$. Now, $\{u_{i_j}\}_{j=1}^\infty$ is a bounded sequence in $W^{1,p}(U)$, so we can employ the Rellich-Kondrachov theorem to get a subsequence $\{u_{i_{j_m}}\}_{m=1}^\infty$ that converges to some $v$ strongly in $L^p(U)$. Then, all that is left to do is show that $v\equiv u$, which I now have no idea how to achieve.

Any help is greatly appreciated. Thank you so much.

Answer 1

The idea is to show that both imply a weaker notion of convergence that still has uniqueness of limits.

For instance $U_m := u_{i_{j_m}} \to v$ in $L^p$ implies $U_m \to v$ in $\mathcal D'(U)$ (distributions) and $U_m \rightharpoonup u$ in $W^{2,p}$ implies $ U_m \to u$ in $\mathcal D'(U)$. By uniqueness of limits in $\mathcal D'(U)$, $u=v$.

You don't have to go all the way to $\mathcal D'(U)$, e.g. $U_m \rightharpoonup u$ in $W^{2,p}$ implies $ U_m \rightharpoonup u$ in $L^p$ which is already enough.

Searching my own answers I find these related posts that also use uniqueness of limits: $L^1$ cannot be embedded into Sobolev $H^{-\frac{n}{2}}(\mathbb{R^n})$ , A question in the proof that Lipchitz continuous functions implies $W^{1,\infty}$. , Boundedness of the inverse (Evans) why the chosen subsequence converge in $L^2(U)$.

Sometimes there are nice estimates you can use instead, see e.g. A step in the proof of Characterization of $W^{1,\infty}$