TITLE: Hopf-Lax formula motivation
QUESTION [5 upvotes]: I'm studying the Hamilton-Jacobi equation by the Evans's "PDE" book and I got a little confused with the way the author got to the definition of the Hopf-Lax formula.
We are considering the classical case, with convex Hamiltonian $H$:
$$\begin{cases}
    u_t + H(D_xu) = 0 \\
    u(x, 0) = g
\end{cases} $$
The first guess, quite understandably, is that there is a direct connection between Hamilton-Jacobi equation and some calculus of variation problem. We try to minimize the action with additional initial value term:
$$\int_{0}^{t}L(\dot{w}(s))ds+g(w(0))$$
Then we set:
$$ u := \inf_{w} \left\{\int_{0}^{t}L(\dot{w}(s))ds+g(w(0))| w(0)=y, w(t)=x\right\}$$
From that point on, everything is quite easy to follow, but my question is: how can we motivate this guess? Evans mentions there would be some justification in Chapter 10, which talks more about viscosity solutions, but I couldn't find anything there.

REPLY [4 votes]: There is another motivation of the Hopf-Lax formula that is often omitted (though it is described in some of Evans' papers). Notice that for any fixed $y,z \in \mathbb{R}^n$, the linear function
$v(x,t) = g(y) + (x-y)\cdot z - tH(z)$
is a solution of your Hamilton-Jacobi equation. This is true because $D_xv(x,t) = z$ and $v_t(x,t)=-H(z) =-H(D_xv(x,t))$. The Hopf-Lax formula is obtained as a two-parameter envelope solution:
$u(x,t) = \inf_{y \in \mathbb{R}^n}\sup_{z \in \mathbb{R}^n} \{ g(y) + (x-y)\cdot z - tH(z)\}.$
Interpreting the inner sup as a Legendre transform puts the formula in the more usual form
$u(x,t) = \inf_{y \in \mathbb{R}^n}\{g(y) - tH^*(\tfrac{x-y}{t}) \},$
which works for convex $H$. By the way, there is another Hopf-Lax formula obtained by simply swapping the inf and sup:
$u(x,t) = \sup_{z \in \mathbb{R}^n}\inf_{y \in \mathbb{R}^n} \{ g(y) + (x-y)\cdot z - tH(z)\}.$
The inner inf can be interpreted as a Legendre transform to obtain
$u(x,t) = \sup_{z \in \mathbb{R}^n}\{x\cdot z - g^*(z) - tH(z)\}.$
This version holds when $g$ is convex and $H$ is allowed to be nonconvex.