Assume $\Omega$ is nice and $\{u_n\}$ is a bounded sequence in $W^{2,p}(\Omega)$ for all $p\in [1,\infty)$, could we find a subsequence that converges strongly $u_{n_k}\to u$ in $\mathrm{C^2}(\Omega)$?
2026-05-05 15:46:27.1777995987
Boundedness in $W^{2,p}(\Omega)$ for all $1\leq p<\infty$
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No, not even if you have boundedness in $C^2(\overline \Omega)$. Since this space is infinite dimensional, closed balls are not compact.