Torus-like solutions for the Landau-de Gennes model. Part III: torus vs split minimizers

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-27 DOI:10.1007/s00526-024-02743-3

Federico Luigi Dipasquale, Vincent Millot, Adriano Pisante

引用次数: 0

Abstract

We study the behaviour of global minimizers of a continuum Landau–de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of \(\mathbb {S}^1\)-equivariant configurations. It is known from our previous paper (Dipasquale et al. in J Funct Anal 286:110314, 2024) that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of torus or of split type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler–Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimization among all competitors.

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Landau-de Gennes 模型的环状解。第三部分：环状解与分裂最小解

我们研究了向列液晶的连续朗道-德-盖尼斯能量函数的全局最小值在与球（向列液滴）差形的三维轴对称域中、以及在一类受限制的 \(\mathbb {S}^1\) -后向构型中的行为。从我们之前的论文（Dipasquale et al. in J Funct Anal 286:110314, 2024）中可以得知，假设有光滑和单轴（例如各向同性）的边界条件以及内部的物理相关规范约束（柳克秀托夫约束），最小化构型要么是环状的，要么是分裂型的。在这里，我们从具有各向同性边界条件的向列液滴出发，展示了如何通过对域或边界数据进行适当变形，使欧拉-拉格朗日方程的奇异（分裂）解或光滑（环状）解（或甚至两者兼而有之）成为能量最小化配置。因此，我们推导出了所有竞争者最小化的对称性破缺结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.