Hopf-Lax type formula for sub- and supersolutions (Q5954407)

The continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation NEWLINE\[NEWLINE\begin{cases} u_t+H(u,Du)= 0\quad \text{for } (x,t) \in\mathbb{R}^n \times\mathbb{R}_+ \\ u(x,0)=u_0(x), \end{cases} \tag{1}NEWLINE\]NEWLINE are studied. Explicit formulas for continuous as well as for the sub- and supersolutions of (1) under the assumption that \(H(s,p)\) is nonincreasing in \(s\) for all \(p\), are obtained. As a consequence of this, the backward problem for the Hamilton-Jacobi equation is solved. Finally, if \(H(u,p)>0\) for \(|p|\neq 0\), then the supersolution becomes a solution of (1).