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 = 0}, and we give a time-integral estimate of its perimeter.

Uniqueness results for nonlocal Hamilton–Jacobi equations

“…Remark 5.6. The above Proposition is related with results obtained by the fourth author in [17] for the eikonal equation with a changing sign velocity.…”