I tried to find the question here, but I couldn't.
I'm a bit puzzled by Sobolev embeddings at the moment.
In my lecture notes, I found the statement
"If $\Omega \subseteq \mathbb{R}^d$ is bounded and open with $\partial \Omega \in C^1$, $1 \leq p < \infty$, $k,m \in \mathbb{N}_0$ with $m > k$, then the embedding
$$ W^{m,p}(\Omega) \hookrightarrow C^k(\overline{\Omega})$$
is compact, if $m – \tfrac{d}{p} > k$."
In my case, I have $k = 0, \, p = 4, \, m = 1, d \leq 3$.
So the first question is:
$1.)$ From this theorem, I could conclude that functions in $W^{1,4}(\Omega)$ can be identified by functions in $C^0(\overline{\Omega})$, right?
Because if I understand it correctly, this would imply, that I can always find a bounded representant in $W^{1,4}(\Omega)$.
But now I am puzzled, because the book where I wanted to refresh my knowledge about Sobolev Spaces says, that it is not possible to have a continous embedding of the form
$$ W^{m,p}(\Omega) \hookrightarrow C^0(\Omega).$$
(The book gives with $d = 2$ the function $u(x) = \log(\log|x|)$ as example, but I didn't understand that completely yet.)
$2.)$ If this statement is true, why does the embedding work if I take the closure of $\Omega$?
Thank you!
Best Answer
Since you have not received an answer yet, I thought I would share my thoughts on this.
The theorem you cite is correct. It´s statement and proof can be found in Adams, Sobolev Spaces, Theorem 6.2 Part III.
Furthermore, for $\Omega \subset R^{n}$ open, bounded by a sufficiently smooth boundary and $mp > n, \; j\ge0$ we have the (compact) imbedding:
$ W^{j+m,p}(\Omega) \subset \subset C_{B}^{j}(\Omega)$ - this is Part II in the above theorem.
I am not familiar with the counterexample you provided, but I can assure you that for $p > n$ and sufficiently regular $\Omega$, we have $ W^{1,p}(\Omega) \rightarrow C^{0,\gamma}(\Omega)$ - the latter being a Holder space.
A bounded set of Holder-continuous functions can always be extended (holder-continuousely) to the boundary of $\Omega$ - this of course is not true in general for continuous functions! We can then use the Arzela-Ascoli-theorem to show compactness in $C^{0}(\bar \Omega)$, because in any such chain of (continuous) imbeddings, it is always sufficient that only one of them is compact for the hole chain to be compact.
I hope this helps a bit, take a look at the section in Adams and you will probably find more answers.