Applications of the Hopf--Lax Formula for ut+H(u,Du)=0

Abstract

The $\g$ (sub)level set of the solution to $w_t+H(\g,D_xw)=0$ is the same as the $\g$ (sub)level set of the solution to $u_t+H(u,D_xu)=0$, and the solution u may be built from w. This result is applied to determining upper and lower bounds for a solution of ut+H1 (u,Du)+H2 (u,Du)=0, with H1 convex and H2 concave, as well as ut+H(u,Du)=0, but with initial data $u(0,x)=g_1(x) \vee g_2(x)$ or $g_1(x) \wedge g_2(x)$, with g1 quasi-convex and g2 quasi-concave. A differential game in $L^{\i}$ is constructed giving a new proof of the Hopf formula.