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Let $\Omega$ be an open subset of $\mathbb{R}^d$. Let $(f_n)_n$ be a sequence in $H^2(\Omega)$. Let $f\in H^2(\Omega)$. Assume that $f_n\rightarrow f$ weakly in $H^1(\Omega)$ and that $D^{\alpha}f_n\rightarrow D^{\alpha}f$ weakly in $L^2(\Omega)$ for every multiindex $\alpha$ with $|\alpha|=2$. Show that there exists a subsequence $(f_{n_j})_j$ that converges in norm to $f$ in $H^1(\Omega)$.

I think that compact inclusion of Sobolev spaces can help. But here I am not allowed to assume any regularity of the boundary of $\Omega$.

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