NONLOCAL FIRST-ORDER HAMILTON-JACOBI EQUATIONS MODELLING DISLOCATIONS DYNAMICS

Guy Barles (1) , Olivier Ley (1, 2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 IRMAR - Institut de Recherche Mathématique de Rennes

We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions on the initial data and the velocity. These results are based on new $L^1$-type estimates on the viscosity solutions of first-order Hamilton-Jacobi Equations appearing in the so-called ``level-sets approach''. Our work is inspired by and simplifies a recent work of Alvarez, Cardaliaguet and Monneau.

Mots clés: Nonlocal Hamilton-Jacobi Equations; dislocation dynamics; nonlocal front propagation; level-set approach; $L^1$-estimates; viscosity solutions; geometrical properties; lower-bound gradient estimate; semiconvexity; viscosity solutions.

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Olivier Ley. NONLOCAL FIRST-ORDER HAMILTON-JACOBI EQUATIONS MODELLING DISLOCATIONS DYNAMICS. Comm. Partial Differential Equations, 2006, 31 (7-9), pp.1191-1208. ⟨10.1080/03605300500361446⟩. ⟨hal-00021694⟩