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 < n. Set
p ∗ := n p n − p . {\displaystyle p^{*}:={\frac {np}{n-p}}.}
Then the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is continuously embedded in the Lp space L p ∗ ( Ω ; R ) {\displaystyle L^{p^{*}}(\Omega ;\mathbb {R} )} and is compactly embedded in L q ( Ω ; R ) {\displaystyle L^{q}(\Omega ;\mathbb {R} )} for every 1 ≤ q < p ∗ {\displaystyle 1\leq q<p^{*}} . In symbols,
W 1 , p ( Ω ) ↪ L p ∗ ( Ω ) {\displaystyle W^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )}
and
W 1 , p ( Ω ) ⊂⊂ L q ( Ω ) for 1 ≤ q < p ∗ . {\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
W k , p ( M ) ⊂ W ℓ , q ( M ) {\displaystyle 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 Ω ‖ L p ( Ω ) ≤ C ‖ ∇ u ‖ L p ( Ω ) {\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
u Ω := 1 meas ( Ω ) ∫ Ω u ( x ) d x {\displaystyle u_{\Omega }:={\frac {1}{\operatorname {meas} (\Omega )}}\int _{\Omega }u(x)\,\mathrm {d} x}
denotes the mean value of u over Ω.
References
Literature Evans, Lawrence C. (2010). Partial Differential Equations (2nd ed.). American Mathematical Society. ISBN 978-0-8218-4974-3. 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 Rellich, Franz (24 January 1930). "Ein Satz über mittlere Konvergenz". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German). 1930: 30–35. JFM 56.0224.02.