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.
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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 > ℓ and k − n/p > ℓ − n/q then the Sobolev embedding
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 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 multifunctions).
The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality, 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.