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I am not sure my question is research type, but I am sure I can find here an answer.

So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:

Theorem 4 (Characterization of $W^{1,\infty}$). Let $U$ be open and bounded, with $\partial U$ of class $C^1$. Then $u: U\to \mathbb{R}$ is Lipschitz continuous iff $u\in W^{1,\infty}(U)$.

Now, I want to adapt this theorem to the case that $U=M$ is a compact manifold like the $n$dimensional torus, i.e its boundary isn't necessarily $C^1$.

How does that change the proof in Evans' book?

I think it only changes step 3 in the proof, other than that the same argument follows also for the torus, am I wrong here?

Thanks.

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  • $\begingroup$ What is the problem here? If there is no boundary, things only get easier! $\endgroup$
    – Michael Renardy
    Commented Jun 5, 2016 at 18:05
  • $\begingroup$ @MichaelRenardy so there's no need to use here step 3, the rest of the argument follows as usual? $\endgroup$
    – Alan
    Commented Jun 5, 2016 at 18:10

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You are right. This theorem and the extension theorem (the step you are referring to in the proof) holds for domains with Lipschitz boundary. You can find a proof of this in Evans and Gariepy's book on measure theory.

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  • $\begingroup$ So in order to adapt the proof to my case with the torus i only need to abandon step 3 in Evans' PDE, correct? $\endgroup$
    – Alan
    Commented Jun 5, 2016 at 17:05
  • $\begingroup$ Thanks, I checked the book you mentioned and it's indeed there. $\endgroup$
    – Alan
    Commented Jun 5, 2016 at 20:34

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