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Stability of half-degree point defect profiles for 2-D nematic liquid crystal
1. | Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185, USA |
2. | Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
3. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.
References:
[1] |
P. Bauman, J. Park and D. Philips,
Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.
doi: 10.1007/s00205-012-0530-7. |
[2] |
P. Biscari and G. G. Peroli,
A hierarchy of defects in biaxial nematics, Commun. Math. Phys, 186 (1997), 381-392.
doi: 10.1007/s002200050113. |
[3] |
G. Canevari,
Biaxiality in the asymptotic analysis of a 2-d Landau-de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var., 21 (2015), 101-137.
doi: 10.1051/cocv/2014025. |
[4] |
P. de Gennes and J. Prost,
The Physics of Liquid Crystals, 2ndedition, Oxford University Press, Oxford, 1995. |
[5] |
G. Di Fratta, J. M. Robbins, V. Slastikov and A. Zarnescu,
Half-integer point defects in the Q-tensor theory of nematic liquid crystals, Journal of Nonlinear Science, 26 (2016), 121-140.
doi: 10.1007/s00332-015-9271-8. |
[6] |
J. Ericksen,
Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.
doi: 10.1007/BF00380413. |
[7] |
D. Golovaty and J. A. Montero,
On minimizers of a Landau-de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., 213 (2014), 447-490.
doi: 10.1007/s00205-014-0731-3. |
[8] |
S. Gustafson and I. M. Sigal,
The stability of magnetic vortices, Commun. Math. Phys., 212 (2000), 257-275.
doi: 10.1007/PL00005526. |
[9] |
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: 10.1007/BF01238933. |
[10] |
F. Hélein,
Minima de la fonctionelle energie libre des cristaux liquides, C. R. Acad. Sci. Paris, 305 (1987), 565-568.
|
[11] |
Y. Hu, Y. Qu and P. Zhang,
On the disclination lines of nematic liquid crystals, Communications in Computational Physics, 19 (2016), 354-379.
doi: 10.4208/cicp.210115.180515a. |
[12] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals, SIAM J. Math. Anal., 46 (2014), 3390-3425.
doi: 10.1137/130948598. |
[13] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 633-673.
doi: 10.1007/s00205-014-0791-4. |
[14] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. I. H. Poincare-AN, 33 (2016), 1131-1152.
doi: 10.1016/j.anihpc.2015.03.007. |
[15] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of point defects of degree $±1/2$ in a two-dimensional nematic liquid crystal model Calculus of Variations and Partial Differential Equations, 55 (2016), 33pp.
doi: 10.1007/s00526-016-1051-2. |
[16] |
M. Kleman and O. D. Lavrentovich,
Topological point defects in nematic liquid crystals, Philosophical Magazine, 86 (2006), 4117-4137.
doi: 10.1080/14786430600593016. |
[17] |
X. Lamy,
Some properties of the nematic radial hedgehog in the Landau-de Gennes theory, J. Math. Anal. Appl., 397 (2013), 586-594.
doi: 10.1016/j.jmaa.2012.08.011. |
[18] |
E. H. Lieb and M. Loss,
Symmetry of the Ginzburg-Landau mimimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.
doi: 10.4310/MRL.1994.v1.n6.a7. |
[19] |
F.-H. Lin and C. Liu,
Static and dynamic theories of liquid crystals, J. Partial Differ. Equ., 14 (2001), 289-330.
|
[20] |
T.-C. Lin,
The stability of the radial solution to the Ginzburg-Landau equation, Commun. PDE, 22 (1997), 619-632.
doi: 10.1080/03605309708821276. |
[21] |
A. Majumdar,
The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals, Euro. J. Appl. Math., 23 (2012), 61-97.
doi: 10.1017/S0956792511000295. |
[22] |
A. Majumdar and A. Zarnescu,
Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.
doi: 10.1007/s00205-009-0249-2. |
[23] |
N. D. Mermin,
The topological theory of defects in ordered media, Rev. Modern Phys., 51 (1979), 591-648.
doi: 10.1103/RevModPhys.51.591. |
[24] |
P. Mironescu,
On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.
doi: 10.1006/jfan.1995.1073. |
[25] |
Manuel de Pino, P. Felmer and M. Kowalczyk,
Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's inequality, IMRN, 30 (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[26] |
R. Rosso and E. G. Virga,
Metastable nematic hedgehogs, J. Phys. A, 29 (1996), 4247-4264.
doi: 10.1088/0305-4470/29/14/041. |
[27] |
G. Toulouse and M. Kleman,
Principles of a classification of defects in ordered media, Journal de Physique Lettres, 37 (1976), 149-151.
doi: 10.1051/jphyslet:01976003706014900. |
show all references
References:
[1] |
P. Bauman, J. Park and D. Philips,
Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.
doi: 10.1007/s00205-012-0530-7. |
[2] |
P. Biscari and G. G. Peroli,
A hierarchy of defects in biaxial nematics, Commun. Math. Phys, 186 (1997), 381-392.
doi: 10.1007/s002200050113. |
[3] |
G. Canevari,
Biaxiality in the asymptotic analysis of a 2-d Landau-de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var., 21 (2015), 101-137.
doi: 10.1051/cocv/2014025. |
[4] |
P. de Gennes and J. Prost,
The Physics of Liquid Crystals, 2ndedition, Oxford University Press, Oxford, 1995. |
[5] |
G. Di Fratta, J. M. Robbins, V. Slastikov and A. Zarnescu,
Half-integer point defects in the Q-tensor theory of nematic liquid crystals, Journal of Nonlinear Science, 26 (2016), 121-140.
doi: 10.1007/s00332-015-9271-8. |
[6] |
J. Ericksen,
Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.
doi: 10.1007/BF00380413. |
[7] |
D. Golovaty and J. A. Montero,
On minimizers of a Landau-de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., 213 (2014), 447-490.
doi: 10.1007/s00205-014-0731-3. |
[8] |
S. Gustafson and I. M. Sigal,
The stability of magnetic vortices, Commun. Math. Phys., 212 (2000), 257-275.
doi: 10.1007/PL00005526. |
[9] |
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: 10.1007/BF01238933. |
[10] |
F. Hélein,
Minima de la fonctionelle energie libre des cristaux liquides, C. R. Acad. Sci. Paris, 305 (1987), 565-568.
|
[11] |
Y. Hu, Y. Qu and P. Zhang,
On the disclination lines of nematic liquid crystals, Communications in Computational Physics, 19 (2016), 354-379.
doi: 10.4208/cicp.210115.180515a. |
[12] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals, SIAM J. Math. Anal., 46 (2014), 3390-3425.
doi: 10.1137/130948598. |
[13] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 633-673.
doi: 10.1007/s00205-014-0791-4. |
[14] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. I. H. Poincare-AN, 33 (2016), 1131-1152.
doi: 10.1016/j.anihpc.2015.03.007. |
[15] |
R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of point defects of degree $±1/2$ in a two-dimensional nematic liquid crystal model Calculus of Variations and Partial Differential Equations, 55 (2016), 33pp.
doi: 10.1007/s00526-016-1051-2. |
[16] |
M. Kleman and O. D. Lavrentovich,
Topological point defects in nematic liquid crystals, Philosophical Magazine, 86 (2006), 4117-4137.
doi: 10.1080/14786430600593016. |
[17] |
X. Lamy,
Some properties of the nematic radial hedgehog in the Landau-de Gennes theory, J. Math. Anal. Appl., 397 (2013), 586-594.
doi: 10.1016/j.jmaa.2012.08.011. |
[18] |
E. H. Lieb and M. Loss,
Symmetry of the Ginzburg-Landau mimimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.
doi: 10.4310/MRL.1994.v1.n6.a7. |
[19] |
F.-H. Lin and C. Liu,
Static and dynamic theories of liquid crystals, J. Partial Differ. Equ., 14 (2001), 289-330.
|
[20] |
T.-C. Lin,
The stability of the radial solution to the Ginzburg-Landau equation, Commun. PDE, 22 (1997), 619-632.
doi: 10.1080/03605309708821276. |
[21] |
A. Majumdar,
The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals, Euro. J. Appl. Math., 23 (2012), 61-97.
doi: 10.1017/S0956792511000295. |
[22] |
A. Majumdar and A. Zarnescu,
Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.
doi: 10.1007/s00205-009-0249-2. |
[23] |
N. D. Mermin,
The topological theory of defects in ordered media, Rev. Modern Phys., 51 (1979), 591-648.
doi: 10.1103/RevModPhys.51.591. |
[24] |
P. Mironescu,
On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.
doi: 10.1006/jfan.1995.1073. |
[25] |
Manuel de Pino, P. Felmer and M. Kowalczyk,
Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's inequality, IMRN, 30 (2004), 1511-1527.
doi: 10.1155/S1073792804133588. |
[26] |
R. Rosso and E. G. Virga,
Metastable nematic hedgehogs, J. Phys. A, 29 (1996), 4247-4264.
doi: 10.1088/0305-4470/29/14/041. |
[27] |
G. Toulouse and M. Kleman,
Principles of a classification of defects in ordered media, Journal de Physique Lettres, 37 (1976), 149-151.
doi: 10.1051/jphyslet:01976003706014900. |
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