Lower-bound Gradient Estimates for
First-Order Hamilton-Jacobi Equations
and Applications to
the Regularity of Propagating Fronts

Olivier Ley

This paper is concerned with first-order time-dependent Hamilton-Jacobi
Equations. Exploiting some ideas of Barron and Jensen, we derive
lower-bound estimates for the gradient of a locally Lipschitz
continuous viscosity solution $u$ of equations with a convex Hamiltonian.
Using these estimates in the context of the level-set approach to front
propagation, we investigate the regularity properties of the propagating
front of $u$, namely $\Gamma_t = \{ x \in \R^n : u(x,t)=0 \}$ for $t \geq 0$.
We show that, contrary to the smooth case, such estimates do not
guarantee, in general, any expected regularity for $\Gamma_t$ even if
$u$ is semiconcave.