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## Rellich-Kondrachov compactness theorem

Multivariate Calculus
Nikos Athanasiou
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Joined: Thu Nov 19, 2015 7:27 pm

### Rellich-Kondrachov compactness theorem

Let $\mathcal{U} \subset \mathbb{R}^d$ be an open bounded domain. Prove that $H^1(\mathcal{U}) \subset \subset L^2(\mathcal{U})$ ,i.e. the inclusion is compact, where $H^1$ is the Sobolev space $W^{1,2}$ and $L^2$ is our usual space on the corresponding domain.

In other words, show that every uniformly bounded sequence in $H^1(\mathcal{U})$ has a subsequence which is convergent in $L^2(\mathcal{U})$.

(Not easy)