The Lax solution to a Hamilton-Jacobi equation and its generalizations: Part 2

It is proved that the function defined by the infimum-based Lax formula (for viscosity solutions) provides a solution almost everywhere in x for each fixed t > 0 to the Hamilton-Jacobi, Cauchy problem u(1) + (1)/(2) parallel todelu parallel to(2) = 0, u(x, 0(+)) = v(x), where the Cauchy data function v is lower semicontinuous on real n-space. In addition, a generalization of the Lax formula is developed for the more inclusive Hamilton-Jacobi equation u(1) + (1)/(2) (parallel todeluparallel to(2) - betauparallel touparallel to(2) + <Jx, x>) = 0, where J is a diagonal, positive-definite matrix. (C) 2003 Elsevier Ltd. All rights reserved.