In [this](https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem) Wikipedia article, the Rellich-Kondrachov theorem says that whenever $M\subset\mathbb{R}^n$ is a compact manifold with $C^1$ boundary then
$W^{k,p}(M)$ embeds compactly in $W^{\ell,q}(M)$ if $k>\ell$ and $k-n/p>\ell-n/q$. Does a similar result follow for manifolds with piecewise $C^1$ boundary such as a rectangle?



Rellich-Kondrachov variant for compact manifold with piecewise $C^1$ boundary?
I quote the following remark from "Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations":



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> **2.35 Remark.** We have given only the main results concerning the theorems of Sobolev and Kondrakov. These theorems are proved for the compact manifolds with Lipschitzian boundary in Aubin [17]..
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> [17] Aubin T.-Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100,
> (1976) 149-173.
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