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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Maria Giovanna Mora
Universita degli Studi di Pavia
Title: The equilibrium measure for a nonlocal dislocation energy

Abstract: In this talk I will discuss the minimum problem for a nonlocal energy, that describes the interactions of a system of positive edge dislocations in the plane. The interaction kernel is given by the sum of the Coulomb potential with an anisotropic term, that makes the potential non-radially symmetric. The purely logarithmic potential has been studied in a variety of contexts (Ginzburg-Landau vortices, Coulomb gases, random matrices, Fekete sets) and it is well known that in this case the equilibrium measure is given by the celebrated circle law. I will show that the presence of the anisotropic term in the potential changes dramatically the nature of the equilibrium measure, which turns out to be supported on the vertical axis and distributed according to Wigner's semi-circle law. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the first one in the case of anisotropic kernels. Moreover, it gives a positive answer to the conjecture that positive dislocations tend to arrange themselves in vertical walls.

Recording: http://mm.math.cmu.edu/recordings/cna/MariaGiovannaMora_small_2017.mp4
Date: Thursday, June 1, 2017
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Leoni