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

Abstract

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 t+ 1 2 || ∇u|| 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 t+ 1 2 (|| ∇u|| 2−βu||u|| 2+〈Jx,x〉)=0, where J is a diagonal, positive-definite matrix.