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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

Apala Majumdar 1,

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.
Keywords: Landau-de Gennes, Ginzburg-Landau, singular set..
Mathematics Subject Classification: 35Qxx, 35Jxx, 35Bx.
Citation: Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure & 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.  Google Scholar

[2]

J. M. Ball and A. Zarnescu, Orientability and energy minimization for liquid crystals, Archive for Rational Mechanics and Analysis, 202 (2011), 493-535. Google Scholar

[3]

F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications," 13. Birkhauser, Boston, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Chen, Dirichlet problem for heat flows of harmonic maps in higher dimensions, Math. Z., 208 (1991), 557-565. doi: 10.1007/BF02571545.  Google Scholar

[9]

Y. Chen and F. H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions, Communications in Analysis and Geometry, 1 (1993), 327-346. Google Scholar

[10]

L. Evans, "Partial Differential Equations," American Mathematical Society, Providence, 1998. Google Scholar

[11]

P. G. De Gennes, "The Physics of Liquid Crystals," Oxford, Clarendon Press, 1974. Google Scholar

[12]

H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969. Google Scholar

[13]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, 224, 2, 1977. Google Scholar

[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.  Google Scholar

[15]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quaislinear Equations of Parabolic Types," American Mathematical Society, Providence, 1968. Google Scholar

[16]

F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

N. J. Mottram and C. Newton, Introduction to Q-tensor theory, University of Strathclyde, Department of Mathematics, Research Report, 10 (2004). Google Scholar

[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.  Google Scholar

[24]

E. B. Priestley, P. J Wojtowicz and P. Sheng, "Introduction to Liquid Crystals," Plenum, New York, 1975. Google Scholar

[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.  Google Scholar

[26]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic mappings, Journal of Differential Geometry, 17 (1982), 307-335. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[29]

E. G. Virga, "Variational Theories for Liquid Crystals," Chapman and Hall, London 1994. Google Scholar

show all references

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.  Google Scholar

[2]

J. M. Ball and A. Zarnescu, Orientability and energy minimization for liquid crystals, Archive for Rational Mechanics and Analysis, 202 (2011), 493-535. Google Scholar

[3]

F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications," 13. Birkhauser, Boston, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Chen, Dirichlet problem for heat flows of harmonic maps in higher dimensions, Math. Z., 208 (1991), 557-565. doi: 10.1007/BF02571545.  Google Scholar

[9]

Y. Chen and F. H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions, Communications in Analysis and Geometry, 1 (1993), 327-346. Google Scholar

[10]

L. Evans, "Partial Differential Equations," American Mathematical Society, Providence, 1998. Google Scholar

[11]

P. G. De Gennes, "The Physics of Liquid Crystals," Oxford, Clarendon Press, 1974. Google Scholar

[12]

H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969. Google Scholar

[13]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, 224, 2, 1977. Google Scholar

[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.  Google Scholar

[15]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quaislinear Equations of Parabolic Types," American Mathematical Society, Providence, 1968. Google Scholar

[16]

F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals, Journal of Partial Differential Equations, 14 (2001), 289-330. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

N. J. Mottram and C. Newton, Introduction to Q-tensor theory, University of Strathclyde, Department of Mathematics, Research Report, 10 (2004). Google Scholar

[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.  Google Scholar

[24]

E. B. Priestley, P. J Wojtowicz and P. Sheng, "Introduction to Liquid Crystals," Plenum, New York, 1975. Google Scholar

[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.  Google Scholar

[26]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic mappings, Journal of Differential Geometry, 17 (1982), 307-335. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[29]

E. G. Virga, "Variational Theories for Liquid Crystals," Chapman and Hall, London 1994. Google Scholar

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