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

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  • Additional Information
    • Contributors:
      Monteil, Antonin
    • Publication Information:
      Preprint
    • Publication Information:
      Society for Industrial & Applied Mathematics (SIAM), 2022.
    • Publication Date:
      2022
    • Abstract:
      We consider $\mathbb{S}^2$-valued maps on a domain $��\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $��$ and horizontal penalization on $\partial��$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincar��-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.
    • File Description:
      application/pdf
    • ISSN:
      1095-7154
      0036-1410
    • Accession Number:
      10.1137/21m143011x
    • Accession Number:
      10.48550/arxiv.2106.15830
    • Rights:
      arXiv Non-Exclusive Distribution
    • Accession Number:
      edsair.doi.dedup.....9b1aa25bcb05585d842e937b31a2b6bf