The Nonconvex Multidimensional Riemann Problem for Hamilton–Jacobi Equations

Abstract

Simple inequalities are presented for the viscosity solution of a Hamilton–Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a “maxmin” equals a “minmax” and thus a representation formula for this problem is then obtained, generalizing the classical Hopf s formulas.