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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?

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The formula tells you that the solution has monotone dependency on the initial data: if the data $g$ is increased pointwise, so is the solution $u$. Therefore, the solution $\tilde u_1$ with initial data $g_1+\sup |g_1-g_2|$ will stay above $u_2$. On the other hand, $\tilde u_1$ is nothing but $u_1+\sup |g_1-g_2|$.