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

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 < p. In symbols,

and

Kondrachov embedding theorem

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

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 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,[2] which states that for u  W1,p(Ω; R) (where Ω satisfies the same hypotheses as above),

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

denotes the mean value of u over Ω.

References

  1. Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.
  2. 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.