Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

Guy Barles (1) , Pierre Cardaliaguet (2) , Olivier Ley (1, 3) , Aurélien Monteillet (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LM - Laboratoire de mathématiques de Brest
3 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: 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; $L^1-$dependence in time.; $L^1-$dependence in time

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Pierre Cardaliaguet, Olivier Ley, Aurélien Monteillet. Uniqueness Results for Nonlocal Hamilton-Jacobi Equations. Journal of Functional Analysis, 2009, 257 (5), pp.1261-1287. ⟨10.1016/j.jfa.2009.04.014⟩. ⟨hal-00264757⟩