Today, let us summarize steps during the standard proof of the Rellich embedding theorem on a bounded domain . This theorem says that
Theorem (Rellich). Suppose that is an open, bounded domain with boundary, and that . Then is compactly embedded in for all .
In particular, for any sequence , there exists a subsequence such that strongly in for some .
In order to prove the Rellich theorem, we need the so-called Arzela-Ascoli theorem.
Lemma (The Arzela-Ascoli theorem). Suppose that , , and is equicontinuous. Then there exists a subsequence uniformly on .
The Arzela-Ascoli theorem is well-known. To prove the Rellich theorem, we shall use the standard mollifiers. To do that, we have to extend a little bit.
Step 1. Assume that is also an open, bounded domain with boundary with . By the Sobolev extension theorem, we can define by with
By the Gagliardo-Nirenberg inequality,
For , let denote the standard mollifiers and set . By choosing sufficiently small, .
Step 2. Since
we can get that
This and the -interpolation inequality imply that
Step 3. Our goal is to employ the Arzela-Ascoli theorem. It is not hard to see the following
Hence there exists a subsequence which converges uniformly on so that
By the triangle inequality, we easily get that
Step 4. By choosing and using the diagonal argument to extract further subsequences, we can arrange to find a subsequence such that
This concludes the proof.
We know discuss a simple application of the idea of the above proof. Assume that the sequence of functions solves the following
where and are smooth functions and uniformly bounded in . We claim that, up to subsequences, strongly in . In other words, for fix , the sequence is always compact in . To see this, we use the standard elliptic regularity theory.
Step 1. Indeed, since are uniformly bounded in , we can see that are uniformly bounded in for any . Since all coefficients of the PDE are uniformly bounded (and even smooth) in , we get via the standard -estimates that for any . Using the Sobolev embedding theorem, the following is compact for some . By the Schauder estimates, we can conclude that .
Step 2. Since all coefficients of the PDE are bounded, we can conclude that the sequence is bounded in .
Step 3. To complete the proof, let say we have a compact subset in . It is obvious to see that the embedding is compact. Indeed, the uniformly boundedness and the equicontinuous come from the following estimates
By the Arzela-Ascoli theorem, there is a subsequence converges uniformly in . In order to use this compact embedding, we need some boundedness of in . Thanks to Step 2, we can conclude that strongly in up to subsequences.