Let $\Omega \subset \mathbb{R}^N$ be open, bounded and $C^1$.
I want to show that $W^{1,\infty}(\Omega) = C^{0,1}(\overline{\Omega})$, with equivalent norms.
I know that $W^{1,\infty}(\Omega) \hookrightarrow C^{0,1}(\overline{\Omega})$ by using Morrey's theorem. Using the characterization theorem by difference quotients (proposition 9.3 in Brezis), I was able to deduce that $C^{0,1}(\Omega) \subset W^{1,\infty}(\Omega)$. However, I'm unable to prove that this injection is continuous. The only issue I'm facing is to prove that $||{\nabla}u||_{L^{\infty}(\Omega)} \leq C [u]_{C^{0,1}(\overline{\Omega})}$, where the right hand side is the Lipschitz semi-norm. Any help is much appreciated.