Minimizing movements for dislocation dynamics with a mean curvature term

¹ CERMICS, École des Ponts, Paris Tech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. forcadel@cermics.enpc.fr
² Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France. aurelien.monteillet@univ-brest.fr

Received: 25 May 2007
Revised: 20 December 2007

Abstract

We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.