Symmetry properties of minimizers of a perturbed Dirichlet energy with a boundary penalization

TL;DR

This paper investigates the symmetry and uniqueness of energy minimizers for a perturbed Dirichlet energy with boundary penalization, with applications to liquid crystals and micromagnetics.

Contribution

It establishes symmetry, uniqueness, and comparison principles for minimizers of a perturbed Dirichlet energy with boundary penalization, extending understanding in liquid crystal and micromagnetic models.

Findings

Minimizers take values in a fixed meridian of the sphere.

Universal constant configurations are globally minimal in certain parameter ranges.

Radial symmetry and monotonicity of minimizers on a ball in 2D.

Abstract

We consider $\mathbb{S}^2$-valued maps on a domain $\Omega\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $\Omega$ and horizontal penalization on $\partial\Omega$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincar\'e-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minimizers with different penalizations. Finally, we apply these results to a problem on a ball and show radial symmetry and monotonicity of minimizers. In dimension $N=2$ our results can be applied to the Oseen--Frank energy for nematic liquid crystals and micromagnetic energy in a thin-film regime.