Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals

TL;DR

This paper analyzes the asymptotic behavior of minimizers in a 2-D Landau-de Gennes model for liquid crystals, showing they are maximally biaxial near singularities and discussing their convergence as elastic constants vanish.

Contribution

It provides a rigorous proof that minimizers are maximally biaxial near singularities and extends the analysis to a general setting beyond the specific Landau-de Gennes problem.

Findings

Minimizers are maximally biaxial near singularities.

Convergence of minimizers as elastic constant approaches zero.

Asymptotic analysis in a general framework.

Abstract

We consider the Landau-de Gennes variational problem on a bound\-ed, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value $1$. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.