Hopf-lax-type formula for ut + H(u, Du) = 0

Abstract

An explicit formula is derived for the Hamilton Jacobi equation ut + H(u, Du) = 0 on (0, ∞) x Rn with u(0, x) = g(x). The hamiltonian H(γ, p) is assumed to be non-decreasing in γ ∈ R1 and convex and positively homogeneous of degree 1 in p ∈ Rn. The unique solution is given by (formula presented) where h is a quasiconvex function given as the quasiconvex dual of H, that is, h(q) = inf{γ ∈ R1 : H(γ, p) ≥ p · q, ∀p}. © 1996 Academic Press, Inc.