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Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Also let

$W^{1,2}(\Omega)=\left\{u \in L^{2}(\Omega)|\,\, \forall \alpha \in \mathbb{N}^{n}:|\alpha| \leq 1\,\, \exists \,D^{\alpha} u \in L^{p}(\Omega)\right\}$

denote the sobolev space.

Futher let $u_n\rightarrow u$ converge weakly in $W^{1,2}(\Omega)$. In the lecture it is sad that this implies $u_n\rightarrow u$ strongly in $L^2(\Omega)$. Does anyone have an explaination?

emily20
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    This is Rellich-Kondrachov Theorem that says that $W^{1,p}$ is compactly embedded in $L^p$. However, your statement is not completely true. It should rather be $u_n\rightharpoonup u$ in $W^{1,p}$ implies $u_n\to u$ in $L^p$ up to a subsequence. I denote $u_n\rightharpoonup u$ for $(u_n)$ converges weakly to $u$ and $u_n\to u$ for $(u_n)$ converges strongly to $u$). – Surb Mar 16 '21 at 20:53

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