Regularity of Minimizers for a General Class of Constrained Energies in Two-Dimensional Domains with Applications to Liquid Crystals

TL;DR

This paper proves regularity properties of minimizers for a broad class of constrained energy functionals in two-dimensional domains, with applications to models of nematic liquid crystals, showing minimizers stay within the interior of the constraint set.

Contribution

It establishes regularity results for minimizers of singular constrained energies, extending understanding in liquid crystal models with boundary constraints.

Findings

Minimizers are regular with finite energy.

Range of minimizers avoids the boundary of the constraint set.

Results apply to modified Landau-de Gennes models.

Abstract

We investigate minimizers defined on a bounded domain $\Omega$ in $\mathbb{R}^2$ for singular constrained energy functionals that include Ball and Majumdar's modification of the Landau-de Gennes Q-tensor model for nematic liquid crystals. We prove regularity of minimizers with finite energy and show that their range on compact subdomains of $\Omega$ does not intersect the boundary of the constraining set.