SOME DIFFERENTIAL PROPERTIES OF A HOPF-TYPE FORMULA FOR HAMILTON - JACOBI EQUATIONS

Abstract

A Hopf-type formula of the Cauchy problem for Hamilton - Jacobi equations (H, σ) is defined by u(t, x) = maxq∈Rn {x, q−σ∗(q)− t 0 H(τ,q)dτ}.
We investigate the points on the domain Ω where the function u(t, x) is differentiable, and the strip of the form (0, t0) × Rn of Ω where the function u(t, x) is continuously differentiable. Moreover, we present a simple propagation of singularity in forward of u(t, x).