Line Defects in the Small Elastic Constant Limit of a Three-Dimensional Landau-de Gennes Model

Canevari, Giacomo

doi:10.1007/s00205-016-1040-9

Cited by 38 publications

(49 citation statements)

References 63 publications

(109 reference statements)

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“…[33,30,20,28,29,25,16,1] and the references therein). In the limit ε → 0, which is referred to as the limit of small nematic correlation length, one recovers the model (2), at least formally; rigorous statements can be found in [33,23,11,12,17].…”

Section: Introductionmentioning

confidence: 71%

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

Canevari

Orlandi

2020

Commun. Math. Phys.

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We consider minimising p-harmonic maps from three-dimensional domains to the real projective plane, for 1 < p < 2. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular set of such a map decomposes into a 1-dimensional set, which can be physically interpreted as a non-orientable line defect, and a locally finite set, i.e. a collection of point defects.

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Section: Introductionmentioning

confidence: 71%

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

Canevari

Orlandi

2020

Commun. Math. Phys.

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“…Generally speaking, for non-orientable boundary conditions on a two-dimensional domain, the Landau-de Gennes energy E ε [Q ε ] of a minimising sequence Q ε diverges logarithmically as ε → 0 (cf. [8]), and an analysis different from the one developed in this paper is required to describe the small-ε behaviour. However, in the special case b 2 = 0 in the Landau-de Gennes bulk potential, results similar to those of Section 2 can be established.…”

Section: The Case B 2 = 0 and Non-orientable Boundary Conditionsmentioning

confidence: 99%

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

Fratta¹,

Robbins²,

Slastikov³

et al. 2020

Arch Rational Mech Anal

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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 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.Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals 2 subject to Dirichlet boundary conditions n| ∂Ω = n b , where the K j 's are material-dependent constants. For mathematical analysis, the one-constant approximation, K 1 = K 2 = K 3 , is often adopted, according to which the Oseen-Frank energy reduces to the Dirichlet energy, with harmonic maps as critical points.One shortcoming of this description is that in certain domains, the director field n is more appropriately represented by an RP 2 -valued map, stemming from the fact that orientations n and −n are physically indistinguishable. In simply-connected domains, a continuous RP 2 -valued map n can be lifted to a continuous S 2 -valued map, in which case we say that n is orientable. However, in non-simply-connected domains, this may not hold, in which case we say that n is non-orientable; see [3] for further discussion, where the notion of orientability is extended to n ∈ W 1,p (Ω, RP 2 ).Another difficulty is the description of defect patterns. These are singularities in the director field, which correspond physically to sharp changes in orientational ordering on a microscopic length scale. It is well known that boundary conditions can force the director field to have singularities. This occurs, for example, when Ω is a three-dimensional domain with boundary homeomorphic to S 2 and the boundary map n b : ∂Ω → S 2 has nonzero degree. In this case, in spite of the singularity, the infimum Oseen-Frank energy is finite. The difficulty is more acute when the boundary data n b : ∂Ω → RP 2 is non-orientable. In this case, the Oseen-Frank energy is

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“…Such an extension Q b ∈ W 1,p (Ω; Q max ) might not exist in case p = 2, due to topological obstructions associated with the manifold Q max (see e.g. [6,Proposition 6]).…”

Section: Remarkmentioning

confidence: 99%

Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

Canevari

Majumdar

Stroffolini

2019

Arch Rational Mech Anal

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Ball in the occasion of his 70th birthday. Abstract We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two-and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.

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Line Defects in the Small Elastic Constant Limit of a Three-Dimensional Landau-de Gennes Model

Cited by 38 publications

References 63 publications

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

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