Let $d \in \mathbb{N}$ and $\Omega$ be a bounded domain of $\mathbb{R}^d$. Consider $m,n,p,q \in \mathbb{N}$ and $T>0$.

Is the space $W^{m,p}([0,T],W^{n,q}(\Omega))$ compactly embedded in any space of continuous functions (such as $C^k([0,T],C^l(\Omega))$ with $k,l\in\mathbb{N}$ ) ?

I guess it is true if $n=m$ and $p=q$, in which case we have

$W^{m,p}([0,T],W^{m,p}(\Omega)) = W^{m,p}(]0,T[\times\Omega)$

and we can use classical compact Sobolev embeddings of Rellich-Kondrachov (right?). However, I did not find any references concerning the general case.

Less general results involving compact embeddings for spaces like $L^p([0,T],W^{n,q}(\Omega))$ or $L^{\infty}([0,T],W^{n,q}(\Omega))$ could be a good start too.

Thanks !


The paper "Compact embeddings of vector-valued Sobolev and Besov spaces" by Amann should probably cover your questions.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.