A Variational Singular Perturbation Problem Motivated by Ericksen’s Model for Nematic Liquid Crystals

Dmitry Golovaty, Itai Shafrir

Research output: Contribution to journalArticlepeer-review

Abstract

We study the asymptotic behavior, when ε→ 0 , of the minimizers {uε}ε>0 for the energy Eε(u)=∫Ω(|∇u|2+(1ε2-1)|∇|u||2),over the class of maps u∈ H1(Ω , R2) satisfying the boundary condition u= g on ∂Ω , where Ω is a smooth, bounded and simply connected domain in R2 and g: ∂Ω → S1 is a smooth boundary data of degree D≥ 1. The motivation comes from a simplified version of the Ericksen model for nematic liquid crystals with variable degree of orientation. We prove convergence (up to a subsequence) of { uε} towards a singular S1–valued harmonic map u, a result that resembles the one obtained in Bethuel et al. (Ginzburg–Landau Vortices, Birkhäuser, 1994) for an analogous problem for the Ginzburg–Landau energy. There are however two striking differences between our result and the one involving the Ginzburg–Landau energy. First, in our problem, the singular limit u may have singularities of, degree strictly larger than one. Second, we find that the principle of “equipartition” holds for the energy of the minimizers, i.e., the contributions of the two terms in Eε(uε) are essentially equal.

Original languageEnglish
Pages (from-to)1009-1063
Number of pages55
JournalArchive for Rational Mechanics and Analysis
Volume241
Issue number2
DOIs
StatePublished - Aug 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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