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,
.