Γ-convergence analysis for discrete topological singularities: The anisotropic triangular lattice and the long range interaction energy

Article type: Research Article

Authors: De Luca, Lucia

Affiliations: Zentrum Mathematik – M7, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany. E-mail: lucia.de.luca@ma.tum.de

Abstract: We consider 2D discrete systems, described by scalar functions and governed by periodic interaction potentials. We focus on anisotropic nearest neighbors interactions in the hexagonal lattice and on isotropic long range interactions in the square lattice. In both these cases, we perform a complete Γ-convergence analysis of the energy induced by a configuration of discrete topological singularities. This analysis allows to prove the existence of many metastable configurations of singularities in the hexagonal lattice.

Keywords: discrete topological singularities, dislocations, XY spin systems, Γ-convergence

DOI: 10.3233/ASY-151334

Journal: Asymptotic Analysis, vol. 96, no. 3-4, pp. 185-221, 2016