Calculus of Variations and Geometric Measure Theory
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Crystal dislocation, nonlocal equations and fractional dynamical systems

Serena Dipierro

created by malchiodi on 20 Sep 2017

22 sep 2017 -- 14:00   [open in google calendar]

Scuola Normale Superiore, Aula Bianchi (Lettere)

Abstract.

We study heteroclinic and multibump orbits for a system of equations driven by a nonlocal operator. Our motivation comes from the study of the atom dislocation function in a periodic crystal, according to the Peierls-Nabarro model. The evolution of the dislocation function can be studied by analytic techniques of fractional Laplace type. At a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior potential. Such potential turns out to be either attractive or repulsive, depending on the mutual orientation of the dislocations, and the attractive potentials generate "particle collisions" in finite time. After the collisions, the system relaxes to the equilibrium exponentially fast, and the associated steady states provide a natural setting for the study of dynamics and chaos in a fractional framework.

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