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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 |
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. |
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. |
[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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