Uniqueness results for nonlocal Hamilton–Jacobi equations

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monteillet, Aurélien

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monteillet, Aurélien (2009), Uniqueness results for nonlocal Hamilton–Jacobi equations, Journal of Functional Analysis, 257, 5, p. 1261-1287. http://dx.doi.org/10.1016/j.jfa.2009.04.014

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00264757/fr/

Date

2009

Journal name

Journal of Functional Analysis

Volume

257

Number

Publisher

Elsevier

Pages

1261-1287

Abstract (EN)

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh–Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.

Subjects / Keywords

Nonlocal Hamilton–Jacobi equations; Dislocation dynamics; Fitzhugh–Nagumo system; Nonlocal front propagation; Level-set approach; Geometrical properties; Lower-bound gradient estimate; Viscosity solutions; Eikonal equation; L1-dependence in time