An extension of the 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.

Theorem (Rellich-Kondrachov). Let be an open, bounded Lipschitz domain, and let . Set

.

Then the Sobolev space is continuously embedded in the space for every and is compactly embedded in for every . In symbols,

and

for .

It is worth noticing from the theory of Sobolev spaces that

.

Therefore, we have the following extension

Theorem (Extension of Rellich-Kondrachov). Let be an open, bounded Lipschitz domain, and let . Set

.

Then we have

for

and

for .

Proof. We first place here the proof of the compactness. In fact, its proof comes from the Rellich-Kondrachov theorem (the case ). Indeed, assume and a bounded sequence in . Our aim is to prove is precompact (its closure is compact or there exists a convergent subsequence) in .

Clearly from the definition of one has

for every and multi-index satisfying . In other words, a bounded sequence in for each multi-index satisfying .

By the Rellich-Kondrachov theorem, there exists a convergent subsequence in (still denoted by ), that means

converges in .

Since there are finite number of multi-indexes satisfying , by finite induction it is possible to select a subsequence for which

converges in for any fixed .

Thus

converges in .

The proof of continuity part is almost the same. What we need here is to derive an estimate to ensure that there is a constant such that

.

Again, by the definition of norms in Sobolev spaces, the Rellich-Kondrachov theorem and a very fundamental inequality, one has

The proof now follows.

Corollary. We always have the following compact embedding

.