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 Lp theorem.

Statement of the theorem
Let Ω ⊆ Rn be an open, bounded Lipschitz domain, and let 1 ≤ p &lt; n. Set


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

Then the Sobolev space W1,p(Ω; R) is continuously embedded in the Lp space Lp ∗ (Ω; R) and is compactly embedded in Lq(Ω; R) for every 1 ≤ q &lt; p∗. In symbols,


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

and


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

Kondrachov embedding theorem
On a compact manifold with $C^{1}$ boundary, the Kondrachov embedding theorem states that if $k > ℓ$ and $k − n/p > ℓ − n/q$ then the Sobolev embedding


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

is completely continuous (compact).

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 W1,p(Ω; R) has a subsequence that converges in Lq(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, which states that for u ∈ W1,p(Ω; R) (where Ω satisfies the same hypotheses as above),


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

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


 * $$u_\Omega := \frac{1}{\operatorname{meas} (\Omega)} \int_\Omega u(x) \, \mathrm{d} x $$

denotes the mean value of u over Ω.

Literature

 * Kondrachov, V. I., On certain properties of functions in the space L p .Dokl. Akad. Nauk SSSR 48, 563–566 (1945).
 * Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. 105. American Mathematical Society. pp. xvi+607. ISBN 978-0-8218-4768-8. MR 2527916. Zbl 1180.46001
 * Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. 105. American Mathematical Society. pp. xvi+607. ISBN 978-0-8218-4768-8. MR 2527916. Zbl 1180.46001