Regularity and the Behavior of Eigenvalues for Minimizers of a Constrained $Q$-tensor Energy for Liquid Crystals

TL;DR

This paper studies minimizers of a constrained Q-tensor energy for liquid crystals, proving their regularity and that eigenvalues stay within physically realistic bounds, advancing understanding of nematic liquid crystal configurations.

Contribution

It establishes regularity of minimizers and eigenvalue bounds for a singular energy model in liquid crystal theory, extending previous models.

Findings

Minimizers are proven to be regular.

Eigenvalues of minimizers are within the physical range.

Results apply to several model problems.

Abstract

We investigate minimizers defined on a bounded domain in $\mathbb{R}^2$ for the Maier--Saupe Q--tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification of the Ginzburg--Landau Q--tensor model, so as to constrain the competing states to have eigenvalues in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we are able to use this regularity to prove that minimizers have eigenvalues strictly within the physical range.