Spirals moving by mean curvature. Part I: a comparison principle

Nicolas Forcadel (1) , Cyril Imbert (1) , Régis Monneau (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this paper, we study the motion of spirals by mean curvature in a (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line attains the surface of a crystal. The main result of this paper is a comparison principle for the corresponding quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the fixed point of spirals is taken into account for the first time (to the best of our knowledge), and second, spirals are studied in the whole space. We also prove that the Cauchy problem is well-posed by using Perron's method.

Mots clés: spirals; motion of interfaces; comparison principle; quasi-linear parabolic equation; viscosity solution

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Spirals moving by mean curvature. Part I: a comparison principle. 2010. ⟨hal-00452241v1⟩