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

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,

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).[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),

\|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.

References

Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.
Evans, Lawrence C. (2010). "§5.8.1". Partial Differential Equations (2nd ed.). p. 290. ISBN 978-0-8218-4974-3.

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.

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[Taylor-1] Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.

[Evans-2] Evans, Lawrence C. (2010). "§5.8.1". Partial Differential Equations (2nd ed.). p. 290. ISBN 978-0-8218-4974-3.