Rellich–Kondrachov theorem

In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L² theorem and Kondrashov the L^p theorem.

Statement of the theorem

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

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

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,

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

and

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

Kondrachov embedding theorem

On a compact manifold with C¹ boundary, the Kondrachov embedding theorem states that if k > ℓ and k − n/p > ℓ − n/q then the Sobolev embedding

${\displaystyle W^{k,p}(M)\subset W^{\ell ,q}(M)}$

is completely continuous (compact).^[1]

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, in the past the result was sometimes referred to as the Rellich–Kondrachov selection theorem, since one "selects" a convergent subsequence. (However, today the customary name is "compactness theorem", whereas "selection theorem" has a precise and quite different meaning, referring to set-valued functions.)

The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality,^[2] which states that for u ∈ W^1,p(Ω; R) (where Ω satisfies the same hypotheses as above),

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

for some constant C depending only on p and the geometry of the domain Ω, where

${\displaystyle u_{\Omega }:={\frac {1}{\operatorname {meas} (\Omega )}}\int _{\Omega }u(x)\,\mathrm {d} x}$

denotes the mean value of u over Ω.