We consider the Landau−de Gennes variational problem on a bounded, 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
Mots-clés : Landau−de Gennes model, Q-tensor, convergence, biaxiality
@article{COCV_2015__21_1_101_0, author = {Canevari, Giacomo}, title = {Biaxiality in the asymptotic analysis of a {2D} {Landau\ensuremath{-}de} {Gennes} model for liquid crystals}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {101--137}, publisher = {EDP-Sciences}, volume = {21}, number = {1}, year = {2015}, doi = {10.1051/cocv/2014025}, mrnumber = {3348417}, zbl = {1311.35209}, language = {en}, url = {https://www.numdam.org/articles/10.1051/cocv/2014025/} }
TY - JOUR AU - Canevari, Giacomo TI - Biaxiality in the asymptotic analysis of a 2D Landau−de Gennes model for liquid crystals JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 101 EP - 137 VL - 21 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2014025/ DO - 10.1051/cocv/2014025 LA - en ID - COCV_2015__21_1_101_0 ER -
%0 Journal Article %A Canevari, Giacomo %T Biaxiality in the asymptotic analysis of a 2D Landau−de Gennes model for liquid crystals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 101-137 %V 21 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2014025/ %R 10.1051/cocv/2014025 %G en %F COCV_2015__21_1_101_0
Canevari, Giacomo. Biaxiality in the asymptotic analysis of a 2D Landau−de Gennes model for liquid crystals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 101-137. doi : 10.1051/cocv/2014025. https://www.numdam.org/articles/10.1051/cocv/2014025/
Biaxial nematic phase in bent-core thermotropic mesogens. Phys. Rev. Lett. 92 (2004) 145506. | DOI
, and ,Nematic liquid crystals: from Maier–Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525 (2010) 1–11. | DOI
and ,Orientable and non–orientable line field models for uniaxial nematic liquid crystals. Mol. Cryst. Liq. Cryst. 495 (2008) 573–585.
and ,F. Bethuel, Variational Methods for Ginzburg−Landau equations, in Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Vol. 1713 of Lect. Notes Math. Springer, Berlin (1999). | MR | Zbl
Asymptotics for the minimization of a Ginzburg–Landau functional. Cal. Var. Partial Differ. Equ. 1 (1993) 123–148. | DOI | MR | Zbl
, and ,F. Bethuel, H. Brezis and F. Hélein, Ginzburg–Landau Vortices. Birkhäuser, Basel and Boston (1994). | MR | Zbl
F. Bethuel and T. Rivière, A minimization problem related to superconductivity. Ann. Institut Henri Poincaré, Anal. Non Lin. (1995) 243–303. | Numdam | MR | Zbl
Partition hypergroups. Amer. J. Math. 6 (1940) 599–612. | DOI | JFM | MR
,Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. Math. Z. 201 (1989) 83–103. | DOI | MR | Zbl
and ,D. Chiron, Étude mathématique de modèles issus de la physique de la matière condensée. PhD thesis, Université Paris VI, France (2004).
On the multigroups of complete conjugate sets of elements of a group. C.R. (Doklady) Acad. Sci. URSS (N.S.) 49 (1946) 315–317. | MR | Zbl
,G. Di Fratta, J.M. Robbins, V. Slastikov and A. Zarnescu, Profiles of point defects in two dimensions in Landau−de Gennes theory. Preprint (2014) arXiv:1403.2566.
On the local instability of radial hedgehog configurations in nematic liquid crystals under Landau–de Gennes free-energy models. Phys. Rev. E 59 (1999) 563–567. | DOI
and ,On minimizers of the Landau–de Gennes energy functional on planar domains. Arch. Ration. Mech. Anal. 213 (2014) 447–490. | DOI | MR | Zbl
and ,Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105 (1986) 547–570. | DOI | MR | Zbl
, and ,Symmetry of uniaxial global Landau–de Gennes minimizers in the theory of nematic liquid crystals. J. Math. Anal. 44 (2012) 3217–3241. | MR | Zbl
and ,Stability of the vortex defect in the Landau-de Gennes theory for nematic liquid crystals. C.R. Math. Acad. Sci. Paris 351 (2013) 533–537. | DOI | MR | Zbl
, , and ,Universal fine structure of nematic hedgehogs. J. Phys. A 34 (2001) 829–838. | DOI | MR | Zbl
and ,Biaxial torus around nematic point defects. Phys. Rew. E 60 (1999) 1858–1866. | DOI
, and ,X. Lamy, Uniaxial symmetry in nematic liquid crystals. Preprint (2014) arXiv:1402.1058. | Numdam | MR
Convergence of minimizers for the
Thermotropic biaxial nematic liquid crystals. Phys. Rev. Lett. 92 (2004) 145505. | DOI
, , and ,Equilibrium order parameters of liquid crystals in the Landau–de Gennes theory. Eur. J. Appl. Math. 21 (2010) 181–203. | DOI | MR | Zbl
,The Landau−de Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond. Arch. Ration. Mech. Anal. 196 (2010) 227–280. | DOI | MR | Zbl
and ,The topological theory of defects in ordered media. Rev. Modern Phys. 51 (1979) 591–648. | DOI | MR | Zbl
,The problem of Plateau on a Riemannian manifold. Ann. Math. 49 (1948) 807–851. | DOI | MR | Zbl
,R. Moser, Partial Regularity for Harmonic Maps and Related Problems. World Scientific Publishing, Singapore (2005). | MR | Zbl
N.J. Mottram and C. Newton, Introduction to Q-tensor theory. Research report, Department of Mathematics, University of Strathclyde (2004).
A. Quarteroni, R. Sacco and F. Saleri, Numer. Math. Springer-Verlag, New York (2000). | MR
The existence of minimal immersions of 2-spheres. Ann. Math. 113 (1981) 1–24. | DOI | MR | Zbl
and ,Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152 (1998) 379–403. Erratum. Ibidem 171 (2000) 233. | DOI | MR | Zbl
,R. Schoen, Analytic aspects of the harmonic map problem, in Seminar on nonlinear partial differential equations, Berkeley, Calif., 1983. Math. Sci. Res. Inst. Publ. Springer, New York (1984). | MR | Zbl
A regularity theory for harmonic maps. J. Differ. Geometry 17 (1982) 307–335. | DOI | MR | Zbl
and ,Defect core structure in nematic liquid crystals. Phys. Rev. 59 (1987) 2582–2584.
and ,Alignment tensor versus director: Description of defects in nematic liquid crystals. Phys. Rev. E 52 (1995) 718–712. | DOI
, and ,On the asymptotic behaviour of the Ginzburg–Landau model in 2 dimensions. J. Differ. Int. Eq. 7 (1994) 1613–1324. Erratum. Ibidem 8 (1995) 224. | MR | Zbl
,- Spherical Particle in Nematic Liquid Crystal with a Magnetic Field and Planar Anchoring, Journal of Nonlinear Science, Volume 35 (2025) no. 1 | DOI:10.1007/s00332-024-10095-7
- On a divergence penalized Landau-de Gennes model, SeMA Journal (2025) | DOI:10.1007/s40324-025-00379-7
- Emergent biaxiality in chiral hybrid liquid crystals, Nature Communications, Volume 15 (2024) no. 1 | DOI:10.1038/s41467-024-54236-8
- The radial hedgehog solution in the Landau–de Gennes theory: Effects of the bulk potentials, Physica D: Nonlinear Phenomena, Volume 459 (2024), p. 134019 | DOI:10.1016/j.physd.2023.134019
- Multistability for Nematic Liquid Crystals in Cuboids with Degenerate Planar Boundary Conditions, SIAM Journal on Applied Mathematics, Volume 84 (2024) no. 2, p. 756 | DOI:10.1137/23m1604606
- Point defects in 2-D liquid crystals with a singular potential: Profiles and stability, Science China Mathematics, Volume 67 (2024) no. 11, p. 2515 | DOI:10.1007/s11425-022-2190-0
- Uniform profile near the point defect of Landau-de Gennes model, Calculus of Variations and Partial Differential Equations, Volume 62 (2023) no. 1 | DOI:10.1007/s00526-022-02348-8
- Pattern formation in Landau–de Gennes theory, Journal of Functional Analysis, Volume 285 (2023) no. 1, p. 109923 | DOI:10.1016/j.jfa.2023.109923
- Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors, Journal of Nonlinear Science, Volume 33 (2023) no. 3 | DOI:10.1007/s00332-023-09898-x
- Singular perturbation of manifold-valued maps with anisotropic energy, Analysis PDE, Volume 15 (2022) no. 6, p. 1531 | DOI:10.2140/apde.2022.15.1531
- Refined asymptotics for Landau-de Gennes minimizers on planar domains, Calculus of Variations and Partial Differential Equations, Volume 61 (2022) no. 6 | DOI:10.1007/s00526-022-02306-4
- Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains, Mathematische Annalen, Volume 383 (2022) no. 3-4, p. 1061 | DOI:10.1007/s00208-021-02204-8
- Modelling and computation of liquid crystals, Acta Numerica, Volume 30 (2021), p. 765 | DOI:10.1017/s0962492921000088
- Topological singularities for vector-valued Sobolev maps and applications, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 30 (2021) no. 2, p. 327 | DOI:10.5802/afst.1677
- Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime, Archive for Rational Mechanics and Analysis, Volume 239 (2021) no. 2, p. 599 | DOI:10.1007/s00205-020-01582-8
- Topological Singular Set of Vector-Valued Maps, II:
-convergence for Ginzburg–Landau type functionals, Archive for Rational Mechanics and Analysis, Volume 241 (2021) no. 2, p. 1065 | DOI:10.1007/s00205-021-01671-2 - The Saturn Ring Effect in Nematic Liquid Crystals with External Field: Effective Energy and Hysteresis, Archive for Rational Mechanics and Analysis, Volume 241 (2021) no. 3, p. 1403 | DOI:10.1007/s00205-021-01674-z
- Ginzburg–Landau Relaxation for Harmonic Maps on Planar Domains into a General Compact Vacuum Manifold, Archive for Rational Mechanics and Analysis, Volume 242 (2021) no. 2, p. 875 | DOI:10.1007/s00205-021-01695-8
- Saturn ring defect around a spherical particle immersed in a nematic liquid crystal, Calculus of Variations and Partial Differential Equations, Volume 60 (2021) no. 6 | DOI:10.1007/s00526-021-02091-6
- Parameter dependent finite element analysis for ferronematics solutions, Computers Mathematics with Applications, Volume 103 (2021), p. 127 | DOI:10.1016/j.camwa.2021.10.027
- Mathematical problems of nematic liquid crystals: between dynamical and stationary problems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 379 (2021) no. 2201, p. 20200432 | DOI:10.1098/rsta.2020.0432
- Disclinations in Limiting Landau–de Gennes Theory, Archive for Rational Mechanics and Analysis, Volume 237 (2020) no. 1, p. 147 | DOI:10.1007/s00205-020-01505-7
- Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals, Archive for Rational Mechanics and Analysis, Volume 237 (2020) no. 3, p. 1421 | DOI:10.1007/s00205-020-01539-x
- Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies, Communications in Mathematical Physics, Volume 374 (2020) no. 3, p. 1483 | DOI:10.1007/s00220-019-03675-2
- Lifting for manifold-valued maps of bounded variation, Journal of Functional Analysis, Volume 278 (2020) no. 10, p. 108453 | DOI:10.1016/j.jfa.2019.108453
- Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy, Archive for Rational Mechanics and Analysis, Volume 233 (2019) no. 3, p. 1169 | DOI:10.1007/s00205-019-01376-7
- Higher dimensional Ginzburg–Landau equations under weak anchoring boundary conditions, Journal of Functional Analysis, Volume 276 (2019) no. 2, p. 447 | DOI:10.1016/j.jfa.2018.07.001
- The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals, Archive for Rational Mechanics and Analysis, Volume 227 (2018) no. 3, p. 1061 | DOI:10.1007/s00205-017-1180-6
- Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime, Journal of Nonlinear Science, Volume 28 (2018) no. 4, p. 1443 | DOI:10.1007/s00332-018-9456-z
- Dimension Reduction for the Landau–de Gennes Model: The Vanishing Nematic Correlation Length Limit, SIAM Journal on Mathematical Analysis, Volume 50 (2018) no. 6, p. 6007 | DOI:10.1137/18m1165189
- Line Defects in the Small Elastic Constant Limit of a Three-Dimensional Landau-de Gennes Model, Archive for Rational Mechanics and Analysis, Volume 223 (2017) no. 2, p. 591 | DOI:10.1007/s00205-016-1040-9
- Uniaxial versus biaxial character of nematic equilibria in three dimensions, Calculus of Variations and Partial Differential Equations, Volume 56 (2017) no. 2 | DOI:10.1007/s00526-017-1142-8
- Stability of half-degree point defect profiles for 2-D nematic liquid crystal, Discrete Continuous Dynamical Systems - A, Volume 37 (2017) no. 12, p. 6227 | DOI:10.3934/dcds.2017269
- Biaxial escape in nematics at low temperature, Journal of Functional Analysis, Volume 272 (2017) no. 10, p. 3987 | DOI:10.1016/j.jfa.2017.01.012
- Dimension Reduction for the Landau-de Gennes Model on Curved Nematic Thin Films, Journal of Nonlinear Science, Volume 27 (2017) no. 6, p. 1905 | DOI:10.1007/s00332-017-9390-5
- Instability of point defects in a two-dimensional nematic liquid crystal model, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 33 (2016) no. 4, p. 1131 | DOI:10.1016/j.anihpc.2015.03.007
- Stability of point defects of degree
± 1 2 in a two-dimensional nematic liquid crystal model, Calculus of Variations and Partial Differential Equations, Volume 55 (2016) no. 5 | DOI:10.1007/s00526-016-1051-2 - Perturbed hedgehogs: continuous deformation of point defects in biaxial nematic liquid crystals, IMA Journal of Applied Mathematics, Volume 81 (2016) no. 4, p. 647 | DOI:10.1093/imamat/hxw005
- Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals, Journal of Nonlinear Science, Volume 26 (2016) no. 1, p. 121 | DOI:10.1007/s00332-015-9271-8
- Liquid crystal defects in the Landau–de Gennes theory in two dimensions — Beyond the one-constant approximation, Mathematical Models and Methods in Applied Sciences, Volume 26 (2016) no. 14, p. 2769 | DOI:10.1142/s0218202516500664
- Radial symmetry on three-dimensional shells in the Landau–de Gennes theory, Physica D: Nonlinear Phenomena, Volume 314 (2016), p. 18 | DOI:10.1016/j.physd.2015.09.013
- A non-traditional view on the modeling of nematic disclination dynamics, Quarterly of Applied Mathematics, Volume 75 (2016) no. 2, p. 309 | DOI:10.1090/qam/1441
- Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films, Journal of Nonlinear Science, Volume 25 (2015) no. 6, p. 1431 | DOI:10.1007/s00332-015-9264-7
- On Minimizers of a Landau–de Gennes Energy Functional on Planar Domains, Archive for Rational Mechanics and Analysis, Volume 213 (2014) no. 2, p. 447 | DOI:10.1007/s00205-014-0731-3
Cité par 44 documents. Sources : Crossref