Hopf-Lax formula and viscosity solution of Hamilton-Jacobi equation

Abstract

Problems involving Hamilton-Jacobi equations arise in many contexts. We will investigate the equation $u_t+H(Du{x,t})$=0 in R^n×(0,∞). First, although classical analysis of the associated problem is limited to local considerations owing to the crossing of characteristics, we derive the Hopf-Lax formula by the calculus of variations and define a weak solution of the Hamilton-Jacobi equation on $R^n×[0, ∞)$ and then we will show the uniqueness of the weak solution under proper assumptions of the Hamiltonian H and the initial data. Next we will investigate the viscosity solution of the Hamiltonian equation. M. G. Crandall and P. L. Lions introduced the notion of viscosity solution of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differential anywhere. The value of this concept is established by the fact that very general existence, uniqueness results hold for viscosity solutions of many problems arising in fields of application. Here we look more closely at the Hopf-Lax formula and two kinds of solutions of the Hamiltonian equation.