Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics.

Francesca da Lio (1) , Nicolas Forcadel (2) , Régis Monneau (2)

1 Dipartimento di Matematica Pura e Applicata [Padova]
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two.

Mots clés: Dislocations dynamics; asymptotic behaviour; non-local equations; eikonal equation; mean curvature motion; viscosity solutions.; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Francesca da Lio, Nicolas Forcadel, Régis Monneau. Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics.. Journal of the European Mathematical Society, 2008, 10 (4), pp.1105--1119. ⟨hal-00415939⟩