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

Giovanni Di Fratta; Alberto Fiorenza; Valeriy Slastikov; Giovanni Di Fratta; Alberto Fiorenza; Valeriy Slastikov

doi:10.3934/mine.2023056

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-38. doi: 10.3934/mine.2023056

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Research article

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

1.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico Ⅱ", Via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy
2.
Dipartimento di Architettura, Università di Napoli Federico Ⅱ, Via Monteoliveto, 3, I-80134 Napoli, Italy
3.
Istituto per le Applicazioni del Calcolo "Mauro Picone", sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino, 111, I-80131 Napoli, Italy
4.
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom

Received: 16 October 2021 Revised: 06 September 2022 Accepted: 02 October 2022 Published: 25 October 2022

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 Poincaré-type inequality on the circular cylinder, which allows for 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.
- Poincaré inequality,
- harmonic maps,
- magnetic skyrmions
Citation: Giovanni Di Fratta, Alberto Fiorenza, Valeriy Slastikov. On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023056

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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 Poincaré-type inequality on the circular cylinder, which allows for 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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