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Analysis of a Ginzburg-Landau type energy model for smectic \(C^\ast\) liquid crystals with defects. (English) Zbl 1288.35443
Summary: This work investigates properties of a smectic \(C^\ast\) liquid crystal film containing defects that cause distinctive spiral patterns in the film’s texture. The phenomena are described by a Ginzburg-Landau type model and the investigation provides a detailed analysis of minimal energy configurations for the film’s director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a renormalized energy. It is shown that if the degree of the boundary data is positive then the vortices each have degree \(+1\) and that they are located away from the boundary. It is proved that the limit of the energies for a sequence of minimizers minus the sum of the energies around their vortices, as the G-L parameter \(\epsilon\) tends to zero, is equal to the renormalized energy for the limiting state.

MSC:
35Q56 Ginzburg-Landau equations
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
76A15 Liquid crystals
49K20 Optimality conditions for problems involving partial differential equations
35R09 Integro-partial differential equations
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[1] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and Their Applications, vol. 13, (1994), Birkhäuser Boston · Zbl 0802.35142
[2] Lin, F., Solutions of Ginzburg-Landau equations and critical points of the renormalized energy, Analyse Non Linéaire, 12, 5, 599-622, (1995) · Zbl 0845.35052
[3] Struwe, M., On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, Differential and Integral Equations, 7, 6, 1613-1624, (1994) · Zbl 0809.35031
[4] Jerrard, R., Lower bounds for generalized Ginzburg-Landau functionals, SIAM Journal on Mathematical Analysis, 30, 4, 721-746, (1999) · Zbl 0928.35045
[5] Sandier, E., Lower bounds for the energy of unit vector fields and applications, Journal of Functional Analysis, 152, 2, 379-403, (1998) · Zbl 0908.58004
[6] Lee, J.-B.; Konovalov, D.; Meyer, R. B., Textural transformations in islands on free standing smectic-C* liquid crystal films, Physical Review E, 73, 051705, (2006), 7 pp
[7] Dacorogna, B., Introduction to the calculus of variations, (2009), Imperial College Press
[8] Lagerwall, S. T., Ferroelectric and antiferroelectric liquid crystals, (1999), Wiley-VCH Weinheim
[9] Meyer, R. B.; Liébert, L.; Strzelecki, L.; Keller, P., Ferroelectric liquid crystals, Le Journal de Physique Lettres, 36, 3, L69-L71, (1975)
[10] Lee, J.-B.; Pelcovits, R. A.; Meyer, R. B., Role of electrostatics in the texture of islands in free-standing ferroelectric liquid crystal films, Physical Review E, 75, 051701, (2007), 5 pp
[11] Kraus, I.; Meyer, R. B., Polar smectic films, Physical Review Letters, 82, 19, 3815-3818, (1999)
[12] Meyer, R. B.; Konovalov, D.; Kraus, I.; Lee, J.-B., Equilibrium size and textures of islands in free-standing smectic C* films, Molecular Crystals and Liquid Crystals, 364, 1, 123-131, (2001)
[13] Silvestre, N.; Patricio, P.; Telo de Gama, M.; Pattanaporkratans, A.; Park, C.; Maclennan, J.; Clark, N., Modeling dipolar and quadrupolar defect structures generated by chiral islands in freely suspended liquid crystal films, Physical Review E, 80, 4, 041708, (2009), 8 pp
[14] Lin, F., Vortex dynamics for the nonlinear wave equation, Communications on Pure and Applied Mathematics, 52, 0737-0761, (1999) · Zbl 0929.35076
[15] Bauman, P.; Park, J.; Phillips, D., Analysis of nematic liquid crystals with disclination lines, Archive for Rational Mechanics and Analysis, 205, 3, 795-826, (2012) · Zbl 1281.76020
[16] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, (1983), Princeton University Press Princeton · Zbl 0516.49003
[17] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (2001), Springer-Verlag Berlin, Heidelberg · Zbl 1042.35002
[18] Lin, F., Static and moving vortices in Ginzburg-Landau theories, Progress in Nonlinear Differential Equations and Their Applications, 29, 71-111, (1997) · Zbl 0867.35039
[19] Lin, F., Some dynamical properties of Ginzburg-Landau vortices, Communications on Pure and Applied Mathematics, 49, 323-359, (1996) · Zbl 0853.35058
[20] Lin, F.; Lin, T.-C., Vortices in p-wave superconductivity, SIAM Journal on Mathematical Analysis, 34, 5, 1105-1127, (2003) · Zbl 1126.82343
[21] Evans, L. C., Weak convergence methods for nonlinear partial differential equations, (Society, A. M., Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, vol. 74, (1990))
[22] Hardt, R.; Kinderlehrer, D.; Lin, F. H., The variety of configurations of static liquid crystals, (Variational Methods, Paris, 1988, Progress in Nonlinear Differential Equations and Their Applications, vol. 4, (1990), Birkhäuser Boston Boston, MA), 115-131
[23] S. Colbert-Kelly, Theoretical and computational analysis of a Ginzburg-Landau type energy model for smectic C* liquid crystals, PhD thesis, Purdue University, 2012.
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