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Consider the initial value problem \partial_t u+ H(Du)=0 \, \, \mbox{in $\mathbb{R}^n \times \mathbb{R}_{>0}$} $ u(0, x)=g(x) \, \, \mbox{on $\mathbb{R}^n \times 0$}$ where $H$ is smooth, convex and coercive and $g$ is Lipschitz. Can you tell me, why the envelope $v(t, x)= \mbox{inf}_{y \in \mathbb{R}^n} \mbox{sup}_{q \in \mathbb{R}^n} g(y)+q \cdot (x-y)- t H(q)$ solves the above initial value problem?

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    The explanation takes several pages in Evans' PDE book, which is a good reference. Is it possible to make the question a little more specific?2011-11-13

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