Abstract
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 W 1,2, 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.
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References
Almgren, F.J., Lieb, E.H.: Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds. Ann. Math. (2) 128(3), 483–530 (1988)
Ball, J.M., Zarnescu, A.: Orientability and energy minimization for liquid crystals (in preparation)
Bethuel F., Brezis H., Hélein F.: Asymptotics for the minimization of a Ginzburg–Landau functional. Calc. Var. Partial Differ. Equ. 1(2), 123–148 (1993)
Bethuel, F., Chiron, D.: Some questions related to the lifting problem in Sobolev spaces. Perspectives in nonlinear partial differential equations, Contemp. Math., 446. Amer. Math. Soc., Providence, RI, 125–152, 2007
Brezis H.: The interplay between analysis and topology in some nonlinear PDE problems. Bull. Am. Math. Soc. (N.S.) 40(2), 179–201 (2003) (electronic)
Chen Y., Lin F.: Remarks on approximate harmonic maps. Comment. Math. Helv. 70(1), 161–169 (1995)
Davis T., Gartland E.: Finite element analysis of the Landau–De Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35, 336–362 (1998)
Ericksen J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1990)
Evans L.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Fraenkel, L.E.: On regularity of the boundary in the theory of Sobolev spaces. Proc. Lond. Math. Soc. (3) 39(3), 385–427 (1979)
Frank F.C.: On the theory of liquid crystals. Disc. Faraday Soc. 25, 1 (1958)
Friedman A.: On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations. J. Math. Mech. 7, 43–59 (1958)
De Gennes P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1974)
Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg, 224, 2, 1977
Hardt R., Lin F.H.: Harmonic maps into round cones and singularities of nematic liquid crystals. Math. Z. 213(4), 575–593 (1993)
Hardt R., Kinderlehrer D., Lin F.H.: Existence and partial regularity of static liquid crystals configurations. Comm. Math. Phys. 105, 547–570 (1986)
Kato T.: Perturbation theory for linear operators, rundlehren der Mathematischen Wissenschaften, Band 132. Springer, Berlin (1976)
Lin F.H.: On nematic liquid crystals with variable degree of orientation. Comm. Pure Appl. Math. 44(4), 453–468 (1991)
Lin F.H., Liu C.: Static and dynamic theories of liquid crystals. J. Partial Differ. Equ. 14(4), 289–330 (2001)
Lin F., Poon C.: On Ericksen’s model for liquid crystals. J. Geom. Anal. 4(3), 379–392 (1994)
Lin F., Riviére T.: Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. (JEMS) 1(3), 237–311 (1999)
Majumdar, A.: Equilibrium order parameters of liquid crystals in the Landau–De Gennes theory, preprint, 2008
De Matteis G., Virga E.G.: Tricritical points in biaxial liquid crystal phases. Phys. Rev. E 71, 061703 (2005)
Mkaddem S., Gartland E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E. 62, 6694–6705 (2000)
Moser R.: Partial Iegularity for Harmonic Maps and Related Problems. World Scientific Publishing, Hackensack (2005)
Mottram, N.J., Newton, C.: Introduction to Q-tensor Theory. University of Strathclyde, Department of Mathematics, Research Report, 10 (2004)
Nomizu K.: Characteristic roots and vectors of a differentiable family of symmetric matrices. Linear Multilinear Algebra 1, 159–162 (1973)
Priestley E.B., Wojtowicz P.J., Sheng P.: Intorduction to Liquid Crystals. Plenum, New York (1975)
Rosso R., Virga E.: Metastable nematic hedgehogs. J. Phys. A Math. Gen. 29, 4247–4264 (1996)
Schoen, R.: Analytic Aspects of the Harmonic Map Problem. Seminar on Nonlinear Partial Differential Equations. (Chern, S.S. Ed. MSRI Publications 2, Springer, Heidelberg, 1984
Schoen R., Uhlenbeck K.: A regularity theory for harmonic mappings. J. Diff. Geom. 17, 307–335 (1982)
Taylor, M.E.: Partial Differential Equations. III. Nonlinear Equations. Applied Mathematical Sciences, 117. Springer, New York, 1997
Virga E.G.: Variational Theories for Liquid Crystals. Chapman and Hall, London (1994)
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Communicated by D. Kinderlehrer
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Majumdar, A., Zarnescu, A. Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond. Arch Rational Mech Anal 196, 227–280 (2010). https://doi.org/10.1007/s00205-009-0249-2
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DOI: https://doi.org/10.1007/s00205-009-0249-2