TY - JOUR
T1 - A Variational Singular Perturbation Problem Motivated by Ericksen’s Model for Nematic Liquid Crystals
AU - Golovaty, Dmitry
AU - Shafrir, Itai
N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.
PY - 2021/8
Y1 - 2021/8
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85107052999&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s00205-021-01670-3
DO - https://doi.org/10.1007/s00205-021-01670-3
M3 - مقالة
SN - 0003-9527
VL - 241
SP - 1009
EP - 1063
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 2
ER -