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arXiv · 2110.08755

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

Giovanni Di Fratta·Alberto Fiorenza·Valeriy Slastikov
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Abstract

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $z$-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincar\'e-type inequality on the circular cylinder, which allows establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.

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Giovanni Di Fratta, Alberto Fiorenza, Valeriy Slastikov. 2021-10-17. On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces. https://arxiv.org/abs/2110.08755

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