On the lax solution to the Hamilton-Jacobi equation. Part I

Abstract

The review article of Crandall, Ishii, and Lions [Bull. AMS,27, No. 1, 1–67 (1992)] devoted to viscosity solutions of first- and second-order partial differential equations contains the exact Lax formula $$u(x,t) = \mathop {\inf }\limits_{y \in R^n } \left\{ {v(y) + \frac{1}{{2t}}\left\| {x - y} \right\|^2 } \right\}$$ ((1)) for a solution to the Hamilton-Jacobi nonlinear partial differential equation $$\frac{{\partial u}}{{\partial t}} + \frac{1}{2}\left\| {\nabla u} \right\|^2 = 0, u\left| {_{t = 0} } \right. = v,$$ ((2)) where the Cauchy datav:R n →R are chosen as a function properly convex and semicontinuous from below, ‖·‖=<·,·> is the usual norm inR n,n ∉Z +, andt ∉R + is a positive evolution parameter. The article also states that there is no exact proof of the Lax formula (1) based on general properties of the Hamiltonian-Jacobi equation (2). This work presents precisely such an exact proof of the Lax formula (1).