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.
Let Ω ⊆ Rn be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set
p*:=
| np | |
| 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,
W1,(\Omega)\hookrightarrow
| p* | |
| L |
(\Omega)
and
W1,(\Omega)\subset\subsetLq(\Omega)for1\leqq<p*.
On a compact manifold with boundary, the Kondrachov embedding theorem states that if and then the Sobolev embedding
Wk,p(M)\subsetW\ell,q(M)
is completely continuous (compact).[1]
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),
\|u-u\Omega
| \| | |
| Lp(\Omega) |
\leqC\|\nablau
| \| | |
| Lp(\Omega) |
for some constant C depending only on p and the geometry of the domain Ω, where
u\Omega:=
| 1 | |
| \operatorname{meas |
(\Omega)}\int\Omegau(x)dx
denotes the mean value of u over Ω.