Rellich-Kondrachov theorem
- Rellich-Kondrachov theorem
In mathematics, the Rellich-Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Italian-Austrian mathematician Franz Rellich.
tatement of the theorem
Let Ω ⊆ R"n" be an open, bounded Lipschitz domain, and let 1 ≤ "p" < "n". Set
:
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,
:
and
:
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, the result is sometimes known as the Rellich-Kondrachov selection theorem (since one "selects" a convergent subsequence).
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),
:
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
*
Wikimedia Foundation.
2010.
Look at other dictionaries:
Sobolev inequality — In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
Continuously embedded — In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are almost equivalent , even though they are not both… … Wikipedia
Espace de Sobolev — Les espaces de Sobolev sont des espaces fonctionnels. Plus précisément, un espace de Sobolev est un espace vectoriel de fonctions muni de la norme obtenue par la combinaison de la norme norme Lp de la fonction elle même ainsi que de ses dérivées… … Wikipédia en Français
