Rellich–Kondrachov theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Italian-Austrian mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrachov the Lp theorem.

Statement of the theorem[edit]

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

p^{*} := \frac{n p}{n - p}.

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,

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

and

W^{1, p} (\Omega) \subset \subset L^{q} (\Omega) \mbox{ for } 1 \leq q < p^{*}.

Consequences[edit]

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, 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,[1] which states that for u ∈ W1,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}{\mathrm{meas} (\Omega)} \int_{\Omega} u(x) \, \mathrm{d} x

denotes the mean value of u over Ω.

References[edit]

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

Literature[edit]

  • Evans, Lawrence C. (2010). Differential Equations, Partial (2nd ed.). American Mathematical Society. ISBN 0-8218-4974-3.