# Is there a Rellich-Kondrachov theorem for manifolds with boundary?

As special case, consider the cylinder $C=[0,T]\times S^n$. Is there a compact embedding $H^1(C)\subset\subset L^2(C)$?

The Wikipedia entry to the Rellich-Kondrachov theorem claims that such an embedding exists for every compact manifold with $C^1$ boundary, but does not give a reference, nor clarifies what is meant by a compact manifold. But this is crucial since most books don't treat manifolds with boundary. Is there a good reference that treats the case I need?

• I don't have references in front of me so can't answer your question, but: as far as I know, there is only one definition of "compact manifold with boundary". – user98602 Nov 11 '15 at 16:47
• I don't have my copy handy to check, but Hebey's book has an appendix on stuff having to do with manifolds with boundary. Maybe he says something about it in there? – Willie Wong Nov 11 '15 at 19:59
• @Willy Wong thank you, I will check that. – tomglabst Nov 11 '15 at 20:11
• You might want to look at Michael Taylor's PDE books. Volume 1 discusses Sobolev spaces (among many other things!). – Phillip Andreae Nov 12 '15 at 5:44
• Thank you Phillip Andreae, Taylor delivers exactly what I was searching for. You might right your comment as an answer. – tomglabst Nov 12 '15 at 9:07

Partial Differential Equations - Basic theory gives an introduction on Sobolev spaces and the main embedding theorems also on compact Riemannian manifolds with smooth boundary, regarded as subsets of compact manifolds without boundary. The searched-for result in the question follows from Rellich's theorem (Prop. 4.4), stating $$H^{s+\sigma}(\Omega)\subset\subset H^s(\Omega)$$ for any $s\geq0$, $\sigma\geq 0$ and $\bar\Omega$ a smooth compact manifold with smooth boundary.