Calculus of Variations and Geometric Measure Theory
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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



Inserted: 29 apr 2021
Last Updated: 29 apr 2021

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 great circle of $\mathbb{S}^2$, and deduce uniqueness under Dirichlet boundary conditions. Finally, we show radial symmetry and monotonicity of minimizers in a ball. 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


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