Abstract
We study the asymptotic behavior, when \(\varepsilon \rightarrow 0\), of the minimizers \(\{u_\varepsilon \}_{\varepsilon >0}\) for the energy
over the class of maps \(u\in H^1(\Omega ,{{\mathbb {R}}}^2)\) satisfying the boundary condition \(u=g\) on \(\partial \Omega \), where \(\Omega \) is a smooth, bounded and simply connected domain in \({{\mathbb {R}}}^2\) and \(g:\partial \Omega \rightarrow S^1\) is a smooth boundary data of degree \(D\ge 1\). 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 \(\{u_\varepsilon \}\) towards a singular \(S^1\)–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 \(E_\varepsilon (u_\varepsilon )\) are essentially equal.
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The research of the first author (DG) was supported in part by the NSF grant DMS-1615952. The research of the second author (IS) was supported by the Israel Science Foundation (Grant No. 894/18). We thank the anonymous referees for their valuable comments and suggestions that helped us to improve the presentation of the manuscript.
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Golovaty, D., Shafrir, I. A Variational Singular Perturbation Problem Motivated by Ericksen’s Model for Nematic Liquid Crystals. Arch Rational Mech Anal 241, 1009–1063 (2021). https://doi.org/10.1007/s00205-021-01670-3
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DOI: https://doi.org/10.1007/s00205-021-01670-3