Biaxiality in the asymptotic analysis of a 2D Landau−de Gennes model for liquid crystals

Canevari, Giacomo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 101-137.

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 $1$. 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.

Classification : 35J25, 35J61, 35B40, 35Q70
Mots clés : Landau−de Gennes model, Q-tensor, convergence, biaxiality

B.R. Acharya, A. Primak and S. Kumar, Biaxial nematic phase in bent-core thermotropic mesogens. Phys. Rev. Lett. 92 (2004) 145506. | DOI

J.M. Ball and A. Majumdar, Nematic liquid crystals: from Maier–Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525 (2010) 1–11. | DOI

J.M. Ball and A. Zarnescu, Orientable and non–orientable line field models for uniaxial nematic liquid crystals. Mol. Cryst. Liq. Cryst. 495 (2008) 573–585.

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

F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg–Landau functional. Cal. Var. Partial Differ. Equ. 1 (1993) 123–148. | DOI | MR | Zbl

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

H. Campaigne, Partition hypergroups. Amer. J. Math. 6 (1940) 599–612. | DOI | JFM | MR

Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. Math. Z. 201 (1989) 83–103. | DOI | MR | Zbl

D. Chiron, Étude mathématique de modèles issus de la physique de la matière condensée. PhD thesis, Université Paris VI, France (2004).

A.P. Dietzman, 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.

E.C. Gartland and S. Mkaddem, 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

D. Golovaty and J.A. Montero, On minimizers of the Landau–de Gennes energy functional on planar domains. Arch. Ration. Mech. Anal. 213 (2014) 447–490. | DOI | MR | Zbl

R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105 (1986) 547–570. | DOI | MR | Zbl

D. Henao and A. Majumdar, Symmetry of uniaxial global Landau–de Gennes minimizers in the theory of nematic liquid crystals. J. Math. Anal. 44 (2012) 3217–3241. | MR | Zbl

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, 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

S. Kralj and E.G. Virga, Universal fine structure of nematic hedgehogs. J. Phys. A 34 (2001) 829–838. | DOI | MR | Zbl

S. Kralj, E.G. Virga and S. Žumer, Biaxial torus around nematic point defects. Phys. Rew. E 60 (1999) 1858–1866. | DOI

X. Lamy, Uniaxial symmetry in nematic liquid crystals. Preprint (2014) arXiv:1402.1058. | Numdam | MR

S. Luckhaus, Convergence of minimizers for the $p$-Dirichlet integral. Math. Z. 213 (1993) 449–456. | DOI | MR | Zbl

L.A. Madsen, T.J. Dingemans, M. Nakata and E.T. Samulski, Thermotropic biaxial nematic liquid crystals. Phys. Rev. Lett. 92 (2004) 145505. | DOI

A. Majumdar, Equilibrium order parameters of liquid crystals in the Landau–de Gennes theory. Eur. J. Appl. Math. 21 (2010) 181–203. | DOI | MR | Zbl

A. Majumdar and A. Zarnescu, 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

N.D. Mermin, The topological theory of defects in ordered media. Rev. Modern Phys. 51 (1979) 591–648. | DOI | MR | Zbl

C.B. Morrey, Jr., 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

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Ann. Math. 113 (1981) 1–24. | DOI | MR | Zbl

E. Sandier, 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

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps. J. Differ. Geometry 17 (1982) 307–335. | DOI | MR | Zbl

N. Schopohl and T.J. Sluckin, Defect core structure in nematic liquid crystals. Phys. Rev. 59 (1987) 2582–2584.

A. Sonnet, A. Killian and S. Hess, Alignment tensor versus director: Description of defects in nematic liquid crystals. Phys. Rev. E 52 (1995) 718–712. | DOI

M. Struwe, 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