Symmetry Properties of Minimizers of a Perturbed Dirichlet Energy with a Boundary Penalization

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 half-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 the micromagnetic energy in a thin-film regime.