### Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts

Olivier Ley

#### Abstract

This paper is concerned with first-order time-dependent Hamilton-Jacobi equations. Exploiting some ideas of Barron and Jensen [9], 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 $\Gamma_t = \{ x \in \mathbb R^n : u(x,t)=0 \}$ 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.

#### Article information

Source
Adv. Differential Equations, Volume 6, Number 5 (2001), 547-576.

Dates
First available in Project Euclid: 2 January 2013

Mathematical Reviews number (MathSciNet)
MR1826721

Zentralblatt MATH identifier
1015.35031

Subjects
Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 35B37 49L25: Viscosity solutions

#### Citation

Ley, Olivier. Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts. Adv. Differential Equations 6 (2001), no. 5, 547--576. https://projecteuclid.org/euclid.ade/1357141855