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Dynamics of $\lambda$-continued fractions and $\beta$-shifts
Partial regularity of minimum energy configurations in ferroelectric liquid crystals
1. | Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seadaemun-gu, Seoul 120-749, South Korea |
2. | Department of Mathematics, Chungnam National University, 99 Daehak-ro, Gung-Dong Yuseong-gu, Daejeon 305-764, South Korea |
References:
[1] |
P. Bauman, J. Park and D. Phillips, Existence of solutions to boundary value problems for smectic liquid crystals, submitted. |
[2] |
P. Bauman and D. Phillips, Analysis and stability of bent core liquid crystal fibers, submitted. |
[3] |
G. Carbou, Regularity for critical points of a nonlocal energy, Calc. Var. Partial Differential Equations, 5 (1997), 409-433. |
[4] |
J. Chen and T. Lubensky, Landau-ginzburg mean-field theory for the nematic to smectic C and nematic to smectic A liquid crystal transistions, Phys. Rev. A, 14 (1976), 1202-1297. |
[5] |
P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon Press, 1993. |
[6] |
F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 25 (1958), 19-28.
doi: 10.1039/df9582500019. |
[7] |
M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Princeton University Press, Princeton, NJ, 1983. |
[8] |
M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations II," Springer, 1998. |
[9] |
R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism, Commun. in Partial Differential Equations, 25 (2000), 1235-1258.
doi: 10.1080/03605300008821549. |
[10] |
R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570.
doi: 10.1007/BF01238933. |
[11] |
S. T. Lagerwall, "Ferroelectric and Antiferroelectric Liquid Crystals," Wiley-VCH, 1999. |
[12] |
F. M. Leslie, I. W. Stewart, T. Carlsson and M. Nakagawa, Equivalent smectic C liquid crystal energies, Continuum Mech. Thermodyn., 3 (1991), 237-250. |
[13] |
I. Lukyanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1998), 574-581.
doi: 10.1103/PhysRevE.57.574. |
[14] |
C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889.
doi: 10.1039/tf9332900883. |
[15] |
M. A. Osipov and S. A. Pikin, Dipolar and quadrupolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5(1995), 1223-1240. |
[16] |
J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals, SIAM J. Appl. Math., 66 (2006), 2107-2126.
doi: 10.1137/050641120. |
[17] |
J. Park, F. Chen and J. Shen, Modeling and simulation of switchings in ferroelectric liquid crystals, Discrete and Cont. Dyn. Syst., 26 (2010), 1419-1440. |
[18] |
R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom., 17 (1982), 307-335. |
[19] |
______, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18 (1983), 253-268. |
[20] |
L. Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps," Birkäuser Verlag, Basel, Boston, Berlin, 1996. |
show all references
References:
[1] |
P. Bauman, J. Park and D. Phillips, Existence of solutions to boundary value problems for smectic liquid crystals, submitted. |
[2] |
P. Bauman and D. Phillips, Analysis and stability of bent core liquid crystal fibers, submitted. |
[3] |
G. Carbou, Regularity for critical points of a nonlocal energy, Calc. Var. Partial Differential Equations, 5 (1997), 409-433. |
[4] |
J. Chen and T. Lubensky, Landau-ginzburg mean-field theory for the nematic to smectic C and nematic to smectic A liquid crystal transistions, Phys. Rev. A, 14 (1976), 1202-1297. |
[5] |
P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon Press, 1993. |
[6] |
F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 25 (1958), 19-28.
doi: 10.1039/df9582500019. |
[7] |
M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Princeton University Press, Princeton, NJ, 1983. |
[8] |
M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations II," Springer, 1998. |
[9] |
R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism, Commun. in Partial Differential Equations, 25 (2000), 1235-1258.
doi: 10.1080/03605300008821549. |
[10] |
R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570.
doi: 10.1007/BF01238933. |
[11] |
S. T. Lagerwall, "Ferroelectric and Antiferroelectric Liquid Crystals," Wiley-VCH, 1999. |
[12] |
F. M. Leslie, I. W. Stewart, T. Carlsson and M. Nakagawa, Equivalent smectic C liquid crystal energies, Continuum Mech. Thermodyn., 3 (1991), 237-250. |
[13] |
I. Lukyanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1998), 574-581.
doi: 10.1103/PhysRevE.57.574. |
[14] |
C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889.
doi: 10.1039/tf9332900883. |
[15] |
M. A. Osipov and S. A. Pikin, Dipolar and quadrupolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5(1995), 1223-1240. |
[16] |
J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals, SIAM J. Appl. Math., 66 (2006), 2107-2126.
doi: 10.1137/050641120. |
[17] |
J. Park, F. Chen and J. Shen, Modeling and simulation of switchings in ferroelectric liquid crystals, Discrete and Cont. Dyn. Syst., 26 (2010), 1419-1440. |
[18] |
R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom., 17 (1982), 307-335. |
[19] |
______, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18 (1983), 253-268. |
[20] |
L. Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps," Birkäuser Verlag, Basel, Boston, Berlin, 1996. |
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