Rellich-Kondrachov theorem

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In mathematics, the Rellich-Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the South-Tyrolian mathematician Franz Rellich.

[edit] Statement of the theorem

Let $\Omega \subseteq \mathbb{R}^{n}$ be an open, bounded Lipschitz domain, and let $1 \leq p < n$ . Set

$p^{*} := \frac{n p}{n - p}.$

Then the Sobolev space $W 1, p (Ω)$ is a subspace of the L^p space $L^{p^{*}} (\Omega)$ and is compactly embedded in $L q (Ω)$ for every $1 \leq q < p^{*}$ . In symbols,

$W^{1, p} (\Omega) \subsetneq L^{p^{*}} (\Omega)$

and

$W^{1, p} (\Omega) \subset \subset L^{q} (\Omega) \mbox{ for } 1 \leq q < p^{*}.$

[edit] Consequences

The Rellich-Kondrachov theorem may be used to prove the Poincaré inequality, which states that for $u \in W^{1, p} (\Omega)$ (where $Ω$ satisfies the same hypotheses as above),