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

By Olivier Ley. Advances in Differential Equations, 2001

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This paper is concerned with rst-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 ? t = fx 2 I R n : u(x; t) = 0g for t 0. We show that, contrary to the smooth case, such estimates do not guarantee, in general, any expected regularity for ? t even if u is semiconcave.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/792f420ce26bacb5da78a5c78e465c7f14fc8e3d

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Some homogenization results for non-coercive Hamilton–Jacobi equations (2006, background): “We end the proof by remarking that we have a Strong Comparison Result for the F equation since F is continuous and just depends on (px, py) and this allows to compare the subsolution u and the supersolution u to get the complete answer (see, for example, Ley[26]).”
On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems (2008, result): “It would be certainly interesting to develop the same kind of results in the context of parabolic first-order viscosity solutions and try to establish links with existing results on the regularity of propagating fronts such as those of Ley (2001) and Barles and Ley (2006).”