# American Institute of Mathematical Sciences

May  2012, 11(3): 1303-1337. doi: 10.3934/cpaa.2012.11.1303

## The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality

 1 Mathematical Institute, University of Oxford, 24-29 St.Giles, Oxford, United Kingdom

Received  November 2010 Revised  November 2011 Published  December 2011

We study small energy solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and three-dimensional domains separately. In the two-dimensional case, we establish the equivalence of the Landau-de Gennes and Ginzburg-Landau theory. In the three-dimensional case, we give a new definition of the defect set based on the normalized energy. In the three-dimensional uniaxial case, we demonstrate the equivalence between the defect set and the isotropic set and prove the $C^{1,\alpha}$-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some $0 < \alpha < 1$, in the vanishing core limit. Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications.
Citation: Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
##### References:
 [1] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Functional Anal., 41 (1981), 135-174. doi: 10.1016/0022-1236(81)90085-9. [2] J. M. Ball and A. Zarnescu, Orientability and energy minimization for liquid crystals, Archive for Rational Mechanics and Analysis, 202 (2011), 493-535. [3] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications," 13. Birkhauser, Boston, 1994. [4] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of Variations and Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614. [5] H. Brezis, J. M. Coron and E. H. Lieb, Harmonic maps with defects, Communications in Mathematical Physics, 107 (1986), 649-705. doi: 10.1007/BF01205490. [6] X. Chen, C. M. Elliott and Q. Tang, Shooting method for vortex solutions of a complex valued Ginzburg-Landau equation, Proceedings of Royal Society of Edinburgh, 124A (1994), 1075-1088. doi: 10.1017/S0308210500030122. [7] Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z., 201 (1989), 83-103. doi: 10.1007/BF01161997. [8] Y. Chen, Dirichlet problem for heat flows of harmonic maps in higher dimensions, Math. Z., 208 (1991), 557-565. doi: 10.1007/BF02571545. [9] Y. Chen and F. H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions, Communications in Analysis and Geometry, 1 (1993), 327-346. [10] L. Evans, "Partial Differential Equations," American Mathematical Society, Providence, 1998. [11] P. G. De Gennes, "The Physics of Liquid Crystals," Oxford, Clarendon Press, 1974. [12] H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969. [13] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, 224, 2, 1977. [14] R. Hardt, D. Kinderlehrer and F. H. Lin, Existence and partial regularity of static liquid crystals configurations, Comm. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933. [15] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quaislinear Equations of Parabolic Types," American Mathematical Society, Providence, 1968. [16] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. [17] A. Majumdar and A. Zarnescu, The Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Archive of Rational Mechanics and Analysis, 196 (2010), 227-280. doi: 10.1007/s00205-009-0249-2. [18] A. Majumdar, Equilibrium order parameters of liquid crystals in the Landau-de Gennes theory, European Journal of Applied Mathematics, 21 (2010), 181-203. doi: 10.1017/S0956792509990210. [19] D. Henao and A. Majumdar, Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystal, submitted to SIAM Journal of Mathematical Analysis. [20] V. Millot and A. Pisante, Symmetry of local minimizers for the three-dimensional Ginzburg-Landau functional, Journal of European Mathematical Society, 12 (2010), 1069-1096. doi: 10.4171/JEMS/223. [21] S. Mkaddem and E. C. Gartland, Fine structure of defects in radial nematic droplets, Physical Review E, 62 (2000), 6694-6705. doi: 10.1103/PhysRevE.62.6694. [22] N. J. Mottram and C. Newton, Introduction to Q-tensor theory, University of Strathclyde, Department of Mathematics, Research Report, 10 (2004). [23] K. Nomizu, Characteristic roots and vectors of a differentiable family of symmetric matrices, Linear and Multilinear Algebra, 1 (1973), 159-162. doi: 10.1080/03081087308817014. [24] E. B. Priestley, P. J Wojtowicz and P. Sheng, "Introduction to Liquid Crystals," Plenum, New York, 1975. [25] R. Rosso and E. Virga, Metastable nematic hedgehogs, J. Phys. A: Math. Gen., 29 (1996), 4247-4264. doi: 10.1088/0305-4470/29/14/041. [26] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic mappings, Journal of Differential Geometry, 17 (1982), 307-335. [27] I. Shafrir, On a class of singular perturbation problems, in"Handbook of Differential Equations, Section: Stationary Partial Differential Equations" (Michel Chipot and Pavol Quittner eds.), Elsevier/North Holland, (2004), 297-383. [28] D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 40 (2002), 2352-2367. doi: 10.1137/S0036142901393814. [29] E. G. Virga, "Variational Theories for Liquid Crystals," Chapman and Hall, London 1994.

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##### References:
 [1] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Functional Anal., 41 (1981), 135-174. doi: 10.1016/0022-1236(81)90085-9. [2] J. M. Ball and A. Zarnescu, Orientability and energy minimization for liquid crystals, Archive for Rational Mechanics and Analysis, 202 (2011), 493-535. [3] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications," 13. Birkhauser, Boston, 1994. [4] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of Variations and Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614. [5] H. Brezis, J. M. Coron and E. H. Lieb, Harmonic maps with defects, Communications in Mathematical Physics, 107 (1986), 649-705. doi: 10.1007/BF01205490. [6] X. Chen, C. M. Elliott and Q. Tang, Shooting method for vortex solutions of a complex valued Ginzburg-Landau equation, Proceedings of Royal Society of Edinburgh, 124A (1994), 1075-1088. doi: 10.1017/S0308210500030122. [7] Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z., 201 (1989), 83-103. doi: 10.1007/BF01161997. [8] Y. Chen, Dirichlet problem for heat flows of harmonic maps in higher dimensions, Math. Z., 208 (1991), 557-565. doi: 10.1007/BF02571545. [9] Y. Chen and F. H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions, Communications in Analysis and Geometry, 1 (1993), 327-346. [10] L. Evans, "Partial Differential Equations," American Mathematical Society, Providence, 1998. [11] P. G. De Gennes, "The Physics of Liquid Crystals," Oxford, Clarendon Press, 1974. [12] H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969. [13] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, 224, 2, 1977. [14] R. Hardt, D. Kinderlehrer and F. H. Lin, Existence and partial regularity of static liquid crystals configurations, Comm. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933. [15] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quaislinear Equations of Parabolic Types," American Mathematical Society, Providence, 1968. [16] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. [17] A. Majumdar and A. Zarnescu, The Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Archive of Rational Mechanics and Analysis, 196 (2010), 227-280. doi: 10.1007/s00205-009-0249-2. [18] A. Majumdar, Equilibrium order parameters of liquid crystals in the Landau-de Gennes theory, European Journal of Applied Mathematics, 21 (2010), 181-203. doi: 10.1017/S0956792509990210. [19] D. Henao and A. Majumdar, Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystal, submitted to SIAM Journal of Mathematical Analysis. [20] V. Millot and A. Pisante, Symmetry of local minimizers for the three-dimensional Ginzburg-Landau functional, Journal of European Mathematical Society, 12 (2010), 1069-1096. doi: 10.4171/JEMS/223. [21] S. Mkaddem and E. C. Gartland, Fine structure of defects in radial nematic droplets, Physical Review E, 62 (2000), 6694-6705. doi: 10.1103/PhysRevE.62.6694. [22] N. J. Mottram and C. Newton, Introduction to Q-tensor theory, University of Strathclyde, Department of Mathematics, Research Report, 10 (2004). [23] K. Nomizu, Characteristic roots and vectors of a differentiable family of symmetric matrices, Linear and Multilinear Algebra, 1 (1973), 159-162. doi: 10.1080/03081087308817014. [24] E. B. Priestley, P. J Wojtowicz and P. Sheng, "Introduction to Liquid Crystals," Plenum, New York, 1975. [25] R. Rosso and E. Virga, Metastable nematic hedgehogs, J. Phys. A: Math. Gen., 29 (1996), 4247-4264. doi: 10.1088/0305-4470/29/14/041. [26] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic mappings, Journal of Differential Geometry, 17 (1982), 307-335. [27] I. Shafrir, On a class of singular perturbation problems, in"Handbook of Differential Equations, Section: Stationary Partial Differential Equations" (Michel Chipot and Pavol Quittner eds.), Elsevier/North Holland, (2004), 297-383. [28] D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 40 (2002), 2352-2367. doi: 10.1137/S0036142901393814. [29] E. G. Virga, "Variational Theories for Liquid Crystals," Chapman and Hall, London 1994.
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