On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

Abstract

This article is a continuation of [J. Math. Sci., 99, No. 5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No. 1, 1–67 (2000)]) $$u\left( {x,t} \right) = \mathop {\inf }\limits_{y \in \mathbb{R}^n } \left\{ {\user1{v}\left( y \right) + \frac{1}{{2t}}\left\| {x - y} \right\|^2 } \right\}$$ for a solution to the Hamilton–Jacobi nonlinear partial differential equation $$\frac{{\partial u}}{{\partial t}} + \frac{1}{2}\left\| {\nabla u} \right\|^2 = 0,u\left| {_{t = 0^ + } } \right. = \user1{v}$$ where the Cauchy data \(\user1{v:}\mathbb{R}^\user1{n} \to \mathbb{R}\) are now a function semicontinuous from below, \(\left\| \cdot \right\| = \left\langle { \cdot , \cdot } \right\rangle\) is the usual norm in \(\mathbb{R}^\user1{n}\), \(n \in \mathbb{Z}_ +\), and \(t \in \mathbb{R}_ + \) is a positive evolution parameter. We proved that the Lax formula solves the Cauchy problem (2) at all points \(x \in \mathbb{R}^n\), \(t \in \mathbb{R}_ + \) fixed save for an exceptional set of points R of the F σ type, having zero Lebesgue measure. In addition, we formulate a similar Lax-type formula without proof for a solution to a new nonlinear equation of the Hamilton–Jacobi-type: $$\frac{{\partial u}}{{\partial t}} + \frac{1}{2}\left\| {\nabla u} \right\|^2 - \frac{{{\beta }u}}{2}\left\| x \right\|^2 + \frac{1}{2}\left\langle {Jx,x} \right\rangle = 0$$ where \(\user1{J:}\mathbb{R}^\user1{n} \to \mathbb{R}^\user1{n}\) is a diagonal positive-definite matrix, mentioned in Part I and having interesting applications in modern mathematical physics.