Hopf-Lax type formula for sub- and supersolutions

Abstract

We study the continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation, $u_t + H(u, D u)=0$ in $\mathbb R^{\,n} \times \mathbb R_+$ with $u(x,0)=u_0(x)$. We obtain explicit formulas for continuous as well as for the sub- and supersolutions. In the latter case, furthermore, if $H(u,p) > 0$ for $|p| \neq 0$ then the supersolution becomes a solution.

Article information

Source
Adv. Differential Equations Volume 5, Number 1-3 (2000), 97-119.

Dates
First available in Project Euclid: 27 December 2012