SYMMETRY PROPERTIES OF MINIMIZERS OF A PERTURBED DIRICHLET ENERGY WITH A BOUNDARY PENALIZATION

We considerS 2-valued maps on a domain Ω C R N minimizing a perturbation of the Dirichlet energy with vertical penalization in Ω and horizontal penalization on Ω . 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 half-meridian of the sphere 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.