Let $u_1,u_2: \mathbb R^n \rightarrow \mathbb R$ be solutions of $\partial_t u_i + H(Du_i) = 0$ with initial conditions $u_i(x,0) = g_i(x)$, with $g_1$ and $g_2$ bounded, $H$ smooth and convex. I am trying to prove the inequality
$\sup |u_1(.,t) - u_2(.,t)| \leq \sup | g_1 - g_2| .$
I know that each $u_i$ is given by the formula
$u(x,t) = \min_{y\in \mathbb R^n} \{t H^*((x-y)/t) + g(y) \}$
where $H^*$ is the Legendre transform of $H$. But, I cannot find a way to deduce the desired inequality from this formula. Any suggestions?