Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$ and $u$ be a uniformly convex function define on $\Omega$. Suppose the gradient of $u$ maps $\Omega$ into a subdomain of $\Omega^*$, i.e. $\omega^*=Du(\Omega)\subset\Omega^*$
We further have that $\Omega^*$ is compactly supported, $\omega^*\subset\subset\Omega^*$.
$u$ is not smooth on $\partial\Omega$, so we define on boundary
$\begin{align} D^+u(y)&=\lim_{y'\notin\bar{\Omega},y'\to y}Du(y')\\ D^-u(y)&=\lim_{y'\in\Omega,y'\to y}Du(y')\end{align}$
Can I deduce from that fact that $\omega^*\subset\subset\Omega^*$ to have $$\gamma\cdot D^+u-\gamma\cdot D^-u\geq C_0>0\quad\text{on}\quad\partial\Omega?$$
for some constant $C_0$, $\gamma$ is the outer unit normal on $\partial\Omega$. How can I make use of the compactly supported assumption?