On Minimizers of a Landau–de Gennes Energy Functional on Planar Domains

Golovaty, Dmitry; Montero, Jose Alberto

doi:10.1007/s00205-014-0731-3

Cited by 48 publications

(53 citation statements)

References 11 publications

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“…As in the Ginzburg-Landau case, topological obstructions may imply the lack of an extension operator W 1−1/k,k (∂Ω, N ) → W 1,k (Ω, N ) (see for instance [10]). As a consequence, minimisers u ε subject to a Dirichlet boundary condition u ε = u bd ∈ W 1−1/k,k (∂Ω, N ) may not satisfy uniform energy bounds with respect to ε. Compactness results in the spirit of the Ginzburg-Landau theory have been shown for minimisers of the Landau-de Gennes functional [49,22,35,23]. However, some points that are understood in the Ginzburg-Landau theory -for instance, a variational characterisation of the singular set of the limit or a description of the problem in terms of Γ-convergence, as in [46,3,4] -are still missing, even for the Landau-de Gennes functional.…”

Section: Background and Motivationmentioning

confidence: 99%

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Canevari

Orlandi

2019

Calc. Var.

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We introduce an operator S on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold N of the Euclidean space R m , and coincides with the distributional Jacobian in case N is a sphere. More precisely, the range of S is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use S to characterise strong limits of smooth, N -valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with N -well potentials.

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Section: Background and Motivationmentioning

confidence: 99%

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Canevari

Orlandi

2019

Calc. Var.

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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: 69%

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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“…However, our arguments hold for any real β. Models with weak anchoring surface terms have been a source of much recent research, for example in [1,8,11,17,23,28]. Next, let us introduce in full detail the problem under consideration.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Dimension Reduction for the Landau--de Gennes Model: The Vanishing Nematic Correlation Length Limit

Novack¹

2018

SIAM J. Math. Anal.

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We study nematic liquid crystalline films within the framework of the Landaude Gennes theory in the limit when both the thickness of the film and the nematic correlation length are vanishingly small compared to the lateral extent of the film. We prove Γ-convergence for a sequence of singularly perturbed functionals with a potential vanishing on a high-dimensional set and a Dirichlet condition imposed on admissible functions. This then allows us to prove the existence of local minimizers of the Landau-de Gennes energy in the spirit of [R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), pp. 69-84] despite the lack of compactness arising from the high-dimensional structure of the wells. The limiting energy consists of leading order perimeter terms, similar to Allen-Cahn models, and lower order terms arising from vortex structures reminiscent of Ginzburg-Landau models.

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On Minimizers of a Landau–de Gennes Energy Functional on Planar Domains

Cited by 48 publications

References 11 publications

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

Dimension Reduction for the Landau--de Gennes Model: The Vanishing Nematic Correlation Length Limit

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