Q:

Questions about Lipschitz continuity and differentiability a.e.

First question. Let $\Omega$ be an open set and $u:\Omega\longrightarrow\mathbb{R}$ be a Lipschitz function in $\Omega$ (in short $u\in\operatorname{Lip}(\Omega)$). I do not understand why $u\in\operatorname{Lip}(\overline{\Omega})$.
Second question. Is it true that if $u\in\operatorname{Lip}(\Omega)$ (and hence by Rademacher Theorem $u$ is differentiable a.e. in $\Omega$), then $\|Du\|_{L^{\infty}(\Omega)}<\infty$?
Thank You

A:

First question: do you know that every Lipschitz function $f$ defined on $\Omega$ has a Lipschitz extension to the whole space? This is true if $\Omega$ is a subset of an arbitrary metric space. You can define the extension $\bar f$ by $$\bar f (x) = \inf_{y \in \Omega} ( f(y) + K d(x,y))$$ where $K$ is the Lipschitz constant of $f$ on $\Omega$.  By continuity, any extension of $f$ is uniquely defined on $\overline{\Omega}$. Thus $f \in \mathrm{Lip}(\Omega)$ implies $f \in \mathrm{Lip} (\overline \Omega)$.
Second question: wherever a Lipschitz function is differentiable each partial derivative is bounded by the Lipschitz constant, so the gradient is essentially bounded.

