# 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 and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the *L*^{2} theorem and Kondrachov the *L*^{p} theorem.

## Statement 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

### Kondrachov embedding theorem

On a compact manifold with *C*^{1} 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 *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 multifunctions).

The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality,^{[1]} 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

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

## Literature

- Evans, Lawrence C. (2010).
*Partial Differential Equations*(2nd ed.). American Mathematical Society. ISBN 0-8218-4974-3. - Rellich, Franz (24 January 1930). "Ein Satz über mittlere Konvergenz".
*Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse*(in German).**1930**: 30–35. Zbl 56.0224.02.