Volume 15, Number 1, January-March 2009
|Page(s)||214 - 244|
|Published online||23 January 2009|
Minimizing movements for dislocation dynamics with a mean curvature term
CERMICS, École des Ponts, Paris Tech, 6 et 8 avenue
Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455
Marne-la-Vallée Cedex 2, France. firstname.lastname@example.org
2 Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France. email@example.com
Revised: 20 December 2007
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.
Mathematics Subject Classification: 53C44 / 49Q15 / 49L25 / 28A75 / 58A25
Key words: Front propagation / non-local equations / dislocation dynamics / mean curvature motion / viscosity solutions / minimizing movements / sets of finite perimeter / currents
© EDP Sciences, SMAI, 2008
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