Integral formulations of the geometric eikonal equation

Abstract

We prove integral formulations of the eikonal equation $u_t=c(x,t)|Du|$, equivalent to the notion of viscosity solution in the framework of the set-theoretic approach to front propagation problems. We apply these integral formulations to investigate the regularity of the front: we prove that under regularity assumptions on the velocity $c$, the front has locally finite perimeter in $\{c\ne 0\}$, and we give a time-integral estimate of its perimeter.