199064410
Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals
We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R . We prove that under a symmetric boundary condition carrying a topological defect of degree $$\frac{k}{2}$$ k 2 for some given even non-zero integer k , there are exactly two minimizers for all large enough R . We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.
73628333
Analysis of Nematic Liquid Crystals with Disclination Lines
We investigate the structure of nematic liquid crystal thin films described by the Landau–de Gennes tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elasticity constant goes to zero in the energy, a limiting uniaxial nematic texture forms with a finite number of defects, all of degree $${\frac{1}{2}}$$ or all of degree $$-{\frac{1}{2}}$$ , corresponding to vertical disclination lines at those locations. We also state a result on the limiting behavior of minimizers of the Chern–Simons–Higgs model without magnetic field that follows from a similar proof.
119025909
On the Disclination Lines of Nematic Liquid Crystals
Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. We implement a spectral method that numerically solves the Landau-de Gennes model with high accuracy, which allows us to study the detailed static structure of defects. We observe five types of defect structures. Among them the 1/2-disclination lines are the most stable structure at low temperature. Inspired by numerical results, we obtain the profile of disclination lines analytically. Moreover, the connection and difference between defect patterns under the Landau-de Gennes model and the Oseen-Frank model are discussed. Finally, four conjectures are made to summarize some important characteristics of defects in the Landau-de Gennes theory. This work is a continuing effort to deepen our understanding on defect patterns in nematic liquid crystals.
52884999
Instability of point defects in a two-dimensional nematic liquid crystal model
We study a class of symmetric critical points in a variational $2D$ Landau - de Gennes model where the state of nematic liquid crystals is described by symmetric traceless $3\times 3$ matrices. These critical points play the role of topological point defects carrying a degree $\frac k 2$ for a nonzero integer $k$. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when $k\neq \pm 1, 0$.
32545126
Liquid crystal defects in the Landau-de Gennes theory in two dimensions - beyond the one-constant approximation
We consider the two-dimensional (2D) Landau–de Gennes energy with several elastic constants, subject to general k-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case k = 2. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parametres. We also numerically study the stability properties of the critical points of the Landau–de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.
17016964
Asymptotics for the minimization of a Ginzburg-Landau functional
AbstractLetΩ ⊂ ℝ2 be a smooth bounded simply connected domain. Consider the functional
$$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$
 on the classHg1={u εH1(Ω; ℂ);u=g on ∂Ω} whereg:∂Ω∂ → ℂ is a prescribed smooth map with ¦g¦=1 on ∂Ω∂ and deg(g, ∂Ω)=0. Let uuε be a minimizer for Eε onHg1. We prove that uε → u0 in
$$C^{1,\alpha } (\bar \Omega )$$
 as ε → 0, where u0 is identified. Moreover
$$\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 $$
.
54755865
Scalings and Limits of Landau-de Gennes Models for liquid crystals: a comment on some Recent analytical Papers
Some recent analytical papers have explored limiting behaviors of Landaude Gennes models for liquid crystals in certain extreme ranges of the model parameters: limits of “vanishing elasticity” (in the language of some of these papers) and “low-temperature limits.” We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrinsic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a “London limit” in the Ginzburg-Landau theory of superconductivity or a “large-body limit” in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-de Gennes free-energy functionals.
119167214
Weak Anchoring for a Two-Dimensional Liquid Crystal
We study the weak anchoring condition for nematic liquid crystals in the context of the Landau-De Gennes model. We restrict our attention to two dimensional samples and to nematic director fields lying in the plane, for which the Landau-De Gennes energy reduces to the Ginzburg--Landau functional, and the weak anchoring condition is realized via a penalized boundary term in the energy. We study the singular limit as the length scale parameter $\varepsilon\to 0$, assuming the weak anchoring parameter $\lambda=\lambda(\varepsilon)\to\infty$ at a prescribed rate. We also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation, for $\lambda(\varepsilon)=K\varepsilon^{-\alpha}$ with $\alpha\in (0,1]$. We show that defects lie on the weak anchoring boundary for $\alpha\in (0,\frac12)$, or for $\alpha=\frac12$ and $K$ small, but they occur inside the bulk domain $\Omega$ for $\alpha>\frac12$ or $\alpha=\frac12$ with $K$ large.
46070148
Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals
We consider the Landau-de Gennes variational problem on a bound\-ed, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value $1$. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.
117443183
Line Defects in the Small Elastic Constant Limit of a Three-Dimensional Landau-de Gennes Model
We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set $${\mathscr{S}_{\rm line}}$$Sline of finite length, such that minimizers converge to a locally harmonic map away from $${\mathscr{S}_{\rm line}}$$Sline. Moreover, $${\mathscr{S}_{\rm line}}$$Sline restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples.
2833544
Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond
We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.
42255368
Radial symmetry on three-dimensional shells in the Landau-de Gennes theory
We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau-de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.
119695444
Stability of the Melting Hedgehog in the Landau–de Gennes Theory of Nematic Liquid Crystals
We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (the so called “melting hedgehog”) in the framework of the Landau–de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary Q-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.
39229141
Uniaxial symmetry in nematic liquid crystals
Abstract Within the Landau–de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. We show that the uniaxial symmetry constraint is very restrictive and can in general not be satisfied, except in very symmetric situations. For one- and two-dimensional configurations, we characterize completely the uniaxial equilibria: they must have constant director. In the three dimensional case we focus on the model problem of a spherical droplet with radial anchoring, and show that any uniaxial equilibrium must be spherically symmetric. It was known before that uniaxiality can sometimes be broken by energy minimizers. Our results shed a new light on this phenomenon: we prove here that in one or two dimensions uniaxial symmetry is always broken, unless the director is constant. Moreover, our results concern all equilibrium configurations, and not merely energy minimizers.
40994718
On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy
We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a three-dimensional spherical domain within the framework of Landau–de Gennes model. Two typical strong anchoring boundary conditions, namely homeotropic and mirror-homeotropic anchoring conditions, are considered. For the homeotropic anchoring, we find three different configurations: uniaxial hedgehog, ring and split-core, in both cases with or without the anisotropic energy. For the mirror-homeotropic anchoring, there are also three analogue solutions: the uniaxial hyperbolic hedgehog, ring and split-core for the isotropic energy case. However, when the anisotropic energy is taken into account, the numerical results and rigorous analysis reveal that the uniaxial hyperbolic hedgehog is no longer a solution. Indeed, we find ring solution only for negative $$L_2$$L2 (the elastic coefficient of the anisotropic energy), while both split-core and ring solutions can be stable minimizers for positive $$L_2$$L2. More precisely, the uniaxial hyperbolic hedgehog for $$L_2=0$$L2=0 bifurcates to a split-core solution when $$L_2$$L2 increases and to a ring solution when $$L_2$$L2 decreases. This example shows that the anisotropic energy may significantly affect the symmetry of point defects with degree $$-1$$-1 whenever it is introduced.
119072452
Phase transitions in nematics: textures with tactoids and disclinations
We demonstrate that a first order isotropic-to-nematic phase transition in liquid crystals can be succesfully modeled within the generalized Landau-de Gennes theory by selecting an appropriate combination of elastic constants. The numerical simulations of the model established in this paper qualitatively reproduce the experimentally observed configurations that include interfaces and topological defects in the nematic phase.
15252155
Harmonic maps with defects
Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation. Extensions of these problems are also solved, e.g. points are replaced by “holes,” ℝ3,S2 is replaced by ℝN,SN−1 or by ℝN, ℝPN−1, the latter being appropriate for the theory of liquid crystals.
2667553
Orientability and Energy Minimization in Liquid Crystal Models
Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor $${Q=s \left(n\otimes n-\frac{1}{3} Id\right)}$$. We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.
119138145
Lifting of $${\mathbb {RP}^{d-1}}$$RPd-1-valued maps in BV and applications to uniaxial Q-tensors. With an appendix on an intrinsic BV-energy for manifold-valued maps
We prove that a BV map with values into the projective space $${\mathbb {RP}^{d-1}}$$RPd-1 has a BV lifting with values into the unit sphere $${{\mathbb {S}}^{d-1}}$$Sd-1 that satisfies an optimal BV-estimate. As an application to liquid crystals, this result is also stated for BV maps with values into the set of uniaxial Q-tensors. In order to quantify BV liftings, we prove an explicit formula for an intrinsic BV-energy of maps with values into any compact smooth manifold.
102354915
Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals
We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of $$\Gamma $$ Γ -development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q -tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy.
119124634
Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7
Abstract For e > 0 , we consider the Ginzburg–Landau functional for R N -valued maps defined in the unit ball B N ⊂ R N with the vortex boundary data x on ∂ B N . In dimensions N ≥ 7 , we prove that, for every e > 0 , there exists a unique global minimizer u e of this problem; moreover, u e is symmetric and of the form u e ( x ) = f e ( | x | ) x | x | for x ∈ B N .
119148104
On the uniqueness of minimisers of Ginzburg-Landau functionals
We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for $\mathbb{R}^n$-valued maps under a suitable convexity assumption on the potential and for $H^{1/2} \cap L^\infty$ boundary data that is non-negative in a fixed direction $e\in \mathbb{S}^{n-1}$. Furthermore, we show that, when minimisers are not unique, the set of minimisers is generated from any of its elements using appropriate orthogonal transformations of $\mathbb{R}^n$. We also prove corresponding results for harmonic maps
16370474
Large solutions for harmonic maps in two dimensions
AbstractWe seek critical points of the functionalE(u)=
$$\mathop \smallint \limits_\Omega$$
|βu|2, where Ω is the unit disk in ℝ2 andu:Ω→S2 satisfies the boundary conditionu=γ on ∂Ω. We prove that if γ is not a constant, thenE has a local minimum which is different from the absolute minimum. We discuss in more details the case where γ(x, y)=(Rx,Ry,
$$\sqrt {1 - R^2 }$$
) andR<1.
55416290
Evolution of harmonic maps with Dirichlet boundary conditions
In this paper we shall study a left over problem concerning the heat flow of harmonic maps on manifolds with boundary. Let (M, g) be a compact smooth m-dimensional Reimannian manifold with nonempty smooth boundary <9M, and let (iV, h) be a compact smooth n-dimensional Reimannian manifold without boundary. We denote M U dM by M. Since (AT, h) can be isometrically embedded into an Euclidean space M, for some k > n, we may view TV as a submanifold of R. In local coordinates on M, the energy of a map u : M —> N ^-> R^ is given by