Partial regularity of minimum energy configurations in ferroelectric liquid crystals

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April 2013, 33(4): 1499-1511. doi: 10.3934/dcds.2013.33.1499

Partial regularity of minimum energy configurations in ferroelectric liquid crystals

Kyungkeun Kang ^1, and Jinhae Park ^2,

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

Received July 2011 Revised September 2012 Published October 2012

Abstract
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Considered here is a system of smectic liquid crystals possessing polarizations described by the Oseen-Frank and Chen-Lubensky energies. We establish partial regularity of minimizers for the governing energy functional using the idea of $(c,\beta)$-almost minimizer introduced in [9].

Keywords: liquid crystal, minimizer, Partial regularity, singularity..

Mathematics Subject Classification: Primary: 35J50, 35J47, 47J0.

Citation: Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499

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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