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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

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 & 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., ().   Google Scholar

[2]

P. Bauman and D. Phillips, Analysis and stability of bent core liquid crystal fibers,, submitted., ().   Google Scholar

[3]

G. Carbou, Regularity for critical points of a nonlocal energy, Calc. Var. Partial Differential Equations, 5 (1997), 409-433.  Google Scholar

[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. Google Scholar

[5]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon Press, 1993. Google Scholar

[6]

F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 25 (1958), 19-28. doi: 10.1039/df9582500019.  Google Scholar

[7]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Princeton University Press, Princeton, NJ, 1983.  Google Scholar

[8]

M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations II," Springer, 1998.  Google Scholar

[9]

R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism, Commun. in Partial Differential Equations, 25 (2000), 1235-1258. doi: 10.1080/03605300008821549.  Google Scholar

[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.  Google Scholar

[11]

S. T. Lagerwall, "Ferroelectric and Antiferroelectric Liquid Crystals," Wiley-VCH, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889. doi: 10.1039/tf9332900883.  Google Scholar

[15]

M. A. Osipov and S. A. Pikin, Dipolar and quadrupolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5(1995), 1223-1240. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom., 17 (1982), 307-335.  Google Scholar

[19]

______, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18 (1983), 253-268.  Google Scholar

[20]

L. Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps," Birkäuser Verlag, Basel, Boston, Berlin, 1996.  Google Scholar

show all references

References:
[1]

P. Bauman, J. Park and D. Phillips, Existence of solutions to boundary value problems for smectic liquid crystals,, submitted., ().   Google Scholar

[2]

P. Bauman and D. Phillips, Analysis and stability of bent core liquid crystal fibers,, submitted., ().   Google Scholar

[3]

G. Carbou, Regularity for critical points of a nonlocal energy, Calc. Var. Partial Differential Equations, 5 (1997), 409-433.  Google Scholar

[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. Google Scholar

[5]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon Press, 1993. Google Scholar

[6]

F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 25 (1958), 19-28. doi: 10.1039/df9582500019.  Google Scholar

[7]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Princeton University Press, Princeton, NJ, 1983.  Google Scholar

[8]

M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations II," Springer, 1998.  Google Scholar

[9]

R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism, Commun. in Partial Differential Equations, 25 (2000), 1235-1258. doi: 10.1080/03605300008821549.  Google Scholar

[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.  Google Scholar

[11]

S. T. Lagerwall, "Ferroelectric and Antiferroelectric Liquid Crystals," Wiley-VCH, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. W. Oseen, The theory of liquid crystals, Trans. Faraday Soc., 29 (1933), 883-889. doi: 10.1039/tf9332900883.  Google Scholar

[15]

M. A. Osipov and S. A. Pikin, Dipolar and quadrupolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5(1995), 1223-1240. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom., 17 (1982), 307-335.  Google Scholar

[19]

______, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18 (1983), 253-268.  Google Scholar

[20]

L. Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps," Birkäuser Verlag, Basel, Boston, Berlin, 1996.  Google Scholar

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