Calculus of Variations and Geometric Measure Theory
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G. Canevari - A. Segatti

Defects in Nematic Shells: a $Γ$-convergence discrete-to-continuum approach

created by canevari on 29 Jan 2021



Inserted: 29 jan 2021

Year: 2016

ArXiv: 1612.07720 PDF


In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $\Gamma$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincar\'e-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.

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