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 Italian-Austrian mathematician Franz Rellich.

[edit] Statement of the theorem

Let Ω ⊆ Rⁿ be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set

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

Then the Sobolev space W^1,p(Ω; R) is continuously embedded in the L^p space L^{p^∗}(Ω; R) and is compactly embedded in L^q(Ω; R) for every 1 ≤ q < p^∗. In symbols,

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

and

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

[edit] Consequences

Since an embedding is compact if and only if the inclusion (identity) operator is a compact operator, the Rellich-Kondrachov theorem implies that any uniformly bounded sequence in W^1,p(Ω; R) has a subsequence that converges in L^q(Ω; R). Stated in this form, the result is sometimes known as the Rellich-Kondrachov selection theorem (since one "selects" a convergent subsequence).

The Rellich-Kondrachov theorem may be used to prove the Poincaré inequality, which states that for u ∈ W^1,p(Ω; R) (where Ω satisfies the same hypotheses as above),

$\| u - u_{\Omega} \|_{L^{p} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)}$