Calculus of Variations and Geometric Measure Theory

G. Di Fratta - A. Monteil - V. Slastikov

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

created by difratta on 29 Apr 2021
modified on 14 Jun 2023


Published Paper

Inserted: 29 apr 2021
Last Updated: 14 jun 2023

Journal: SIAM Journal on Mathematical Analysis
Year: 2021


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