Asymptotics for Minimizers of Landau-de Gennes with a Magnetic Field and Tangential Anchoring

TL;DR

This paper establishes the existence and asymptotic behavior of minimizers for the Landau-de Gennes energy in liquid crystals under magnetic fields and anchoring, generalizing previous results and characterizing particle orientations.

Contribution

It proves existence of minimizers, analyzes their asymptotics for arbitrary particle shapes, and characterizes optimal particle orientations relative to magnetic fields.

Findings

No line singularities in certain regimes.

Explicit orientation characterization for various shapes.

Generalization to arbitrary particle geometries.

Abstract

In this article we prove existence of minimizers of the Landau-de Gennes energy for liquid crystals with homogeneous external magnetic field and strong uniaxial planar anchoring. Next we consider the asymptotics of solutions to the joint minimization of the energy w.r.t. the function and its boundary condition. This constitutes a generalization to arbitrary regular particle shapes of the results obtained in [BLS, arXiv:2403.20274] in a particular setting. Moreover, we show the absence of line singularities in some asymptotic parameter regime. Finally we characterize the optimal orientation of particles vis-\`a-vis the magnetic field direction and compute it explicitly for different particle shapes.