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

created by difratta on 29 Apr 2021
modified on 29 Jun 2021

[BibTeX]

Preprint

Inserted: 29 apr 2021
Last Updated: 29 jun 2021

Year: 2021

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 Poincaré-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.

Keywords: harmonic maps, symmetry, Boundary Penalization