Rellich–Kondrachov theorem

Template:Short description 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,

$W^{1,p}(\mathrm{\Omega })\hookrightarrow L^{p^{*}}(\mathrm{\Omega })$

and

$W^{1,p}(\mathrm{\Omega })\subset \subset L^q(\mathrm{\Omega })\text{ for }1\le q<p^{*}.$

Kondrachov embedding theorem

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

$W^{k,p}(M)\subset W^{\ell ,q}(M)$

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 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),

$\Vert u-u_{\mathrm{\Omega }}{\Vert }_{L^p(\mathrm{\Omega })}\le C\Vert \mathrm{\nabla }u{\Vert }_{L^p(\mathrm{\Omega })}$

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

$u_{\mathrm{\Omega }}:=\frac{1}{\mathrm{meas}(\mathrm{\Omega })}\underset{\mathrm{\Omega }}{\int }u(x)\mathrm{d}x$

denotes the mean value of u over Ω.

Literature

• Template:Cite book
• 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. Template:ISBN. MR 2527916. Zbl 1180.46001
• Template:Cite journal