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 ''L''² theorem and Kondrashov the ''L''^''p'' theorem.

Statement of the theorem

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

:$p^ := \frac.$

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,

:$W^ (\Omega) \hookrightarrow L^ (\Omega)$

and

:$W^ (\Omega) \subset \subset L^ (\Omega) \text 1 \leq q < p^.$

Kondrachov embedding theorem

On a compact manifold with boundary, the Kondrachov embedding theorem states that if and then the Sobolev embedding

:$W^(M)\subset W^(M)$

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 ''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, which states that for ''u'' ∈ ''W''^1,''p''(Ω; R) (where Ω satisfies the same hypotheses as above),

:$\, u - u_\Omega \, _ \leq C \, \nabla u \, _$

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

:$u_\Omega := \frac \int_\Omega u(x) \, \mathrm x$

denotes the mean value of ''u'' over Ω.

References

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. . MRbr>2527916
Zblbr>1180.46001
*

{{DEFAULTSORT:Rellich-Kondrachov theorem
Theorems in mathematical analysis
Sobolev spaces