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.



Lipschitz functions and $W^{1,\infty}$
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.


