Geodesic
Uniqueness and existence of spirals moving by forced mean curvature motion
Nicolas Forcadel1 ; Cyril Imbert2 ; Régis Monneau3
1Université Paris-Dauphine, France
2Université Paris-Est Créteil Val de Marne, France
3CERMICS - ENPC, Marne-La-Vallée, France
Interfaces and free boundaries, Tome 14 (2012) no. 3, pp. 365-400

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DOI

In this paper, we study the motion of spirals by mean curvature type motion in 1 the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic 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 attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.
DOI : 10.4171/ifb/285
Classification : 35-XX, 00-XX
Mots-clés : Spirals, motion of interfaces, comparison principle, quasi-linear parabolic equation, viscosity solutions, mean curvature motion

Nicolas Forcadel  1   ; Cyril Imbert  2   ; Régis Monneau  3

1 Université Paris-Dauphine, France
2 Université Paris-Est Créteil Val de Marne, France
3 CERMICS - ENPC, Marne-La-Vallée, France 
Nicolas Forcadel; Cyril Imbert; Régis Monneau. Uniqueness and existence of spirals moving by forced mean curvature motion. Interfaces and free boundaries, Tome 14 (2012) no. 3, pp. 365-400. doi: 10.4171/ifb/285
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     title = {Uniqueness and existence of spirals moving by forced mean curvature motion},
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     pages = {365--400},
     year = {2012},
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     number = {3},
     doi = {10.4171/ifb/285},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/285/}
}
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