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The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals

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Abstract

We study the relationship between Onsager’s molecular theory, which involves the effects of nonlocal molecular interactions and the Oseen–Frank theory for nematic liquid crystals. Under the molecular setting, we prove the existence of global minimizers for the generalized Onsager’s free energy, subject to a nonlocal boundary condition which prescribes the second moment of the number density function near the boundary. Moreover, when the re-scaled interaction distance tends to zero, the global minimizers will converge to a uniaxial distribution predicted by a minimizing harmonic map. This is achieved through the investigations of the compactness property and the boundary behaviors of the corresponding second moments. A similar result is established for critical points of the free energy that fulfill a natural energy bound.

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Authors and Affiliations

  1. NYU Shanghai, 1555 Century Avenue, Shanghai, 200122, China

    Yuning Liu

  2. NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

    Yuning Liu

  3. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Wei Wang

Authors
  1. Yuning Liu
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  2. Wei Wang
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Correspondence to Wei Wang.

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Communicated by D. Kinderlehrer

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Liu, Y., Wang, W. The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals. Arch Rational Mech Anal 227, 1061–1090 (2018). https://doi.org/10.1007/s00205-017-1180-6

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  • DOI: https://doi.org/10.1007/s00205-017-1180-6

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