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?

Solution 1:

Since the answer was given by Willy Wong and Phillip Andreae as comments, I just recap their inputs, giving exact references to the results in the mentioned books.

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.

The appendix of Nonlinear Analysis on Manifolds - Sobolev Spaces and Inequalities by Emmanuel Hebey gives also some results for smooth compact Riemannian manifolds with boundaries, giving an embedding theorem (Thm. 10.1) that implies the classical Rellich Kondrachov theorem. Hebey does not specify the smoothness requirements for the boundary, but refers to Aubin's book.

So finally, Nonlinear Analysis on Manifolds. Monge-Ampère Equations gives the desired reference for the result found on Wikipedia.