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The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition

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Abstract

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space \(H^{3/2}\). In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.

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Authors and Affiliations

  1. Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Bizkaia, Spain

    Zhiyuan Geng

  2. Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Fanghua Lin

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  1. Zhiyuan Geng
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  2. Fanghua Lin
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Correspondence to Zhiyuan Geng.

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Communicated by E. Virga.

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The authors are partially supported by the NSF Grant DMS1955249. The first author is partially supported by the Basque Government through the BERC 2018-2021 program and by the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project PID2020-114189RB-I00 funded by Agencia Estatal de Investigación (PID2020-114189RB-I00/AEI/10.13039/501100011033).

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Geng, Z., Lin, F. The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition. Arch Rational Mech Anal 243, 1181–1221 (2022). https://doi.org/10.1007/s00205-021-01733-5

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  • DOI: https://doi.org/10.1007/s00205-021-01733-5

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