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

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.

• What is the problem here? If there is no boundary, things only get easier! – Michael Renardy Jun 5 '16 at 18:05
• @MichaelRenardy so there's no need to use here step 3, the rest of the argument follows as usual? – Alan Jun 5 '16 at 18:10