Short Time Uniqueness Results for Solutions of Nonlocal and Non-monotone Geometric Equations

Guy Barles (1, 2) , Olivier Ley (3) , Hiroyoshi Mitake (1, 4)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Department of Applied Mathematics

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh-Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.

Mots clés: Nonlocal Hamilton-Jacobi Equations; Nonlocal Front Propagation; Short Time Uniqueness; Non-Fattening Condition; Lower Gradient Estimate; Dislocation Dynamics; Fitzhugh-Nagumo System; Viscosity Solution

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Olivier Ley, Hiroyoshi Mitake. Short Time Uniqueness Results for Solutions of Nonlocal and Non-monotone Geometric Equations. Mathematische Annalen, 2012, 352 (2), pp.409-451. ⟨10.1007/s00208-011-0648-1⟩. ⟨hal-00486802⟩