×

zbMATH — the first resource for mathematics

A reduced study for nematic equilibria on two-dimensional polygons. (English) Zbl 1446.35204

MSC:
35Q82 PDEs in connection with statistical mechanics
76A15 Liquid crystals
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
35J20 Variational methods for second-order elliptic equations
35B32 Bifurcations in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, MA, 1974.
[2] F. Bethuel, H. Brezis, and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), pp. 123-148. · Zbl 0834.35014
[3] F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau Vortices, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1994.
[4] K. Bisht, Y. Wang, V. Banerjee, and A. Majumdar, Tailored morhpologies in two-dimensional ferronematic wells, Phys. Rev. E, 101 (2020), 022706.
[5] M. A. Brilleslyper, M. J. Dorff, J. M. McDougall, J. S. Rolf, L. E. Schaubroek, R. L. Stankewitz, and K. Stephenson, Explorations in Complex Analysis, Mathematical Association of America, Washington, DC, 2012. · Zbl 1253.30003
[6] Y. Cai, P. Zhang, and A.-C. Shi, Liquid crystalline bilayers self-assembled from rod-coil diblock copolymers, Soft Matter, 13 (2017), pp. 4607-4615.
[7] G. Canevari, J. Harris, A. Majumdar, and Y. Wang, The well order reconstruction solution for three-dimensional wells, in the Landau-de Gennes theory, Int. J. Non-Linear Mech., 119 (2020), 103342.
[8] G. Canevari, A. Majumdar, and A. Spicer, Order reconstruction for nematics on squares and hexagons: A Landau-de Gennes study, SIAM J. Appl. Math., 77 (2017), pp. 267-293. · Zbl 1371.35065
[9] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Internat. Ser. Monogr. Phys., 83, Oxford University Press, Oxford, 1995.
[10] T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, Cambridge Monogr. Appl. Comput. Math. 8, Cambridge University Press, Cambridge, 2002. · Zbl 1003.30005
[11] L. Fang, A. Majumdar, and L. Zhang, Surface, size and topological effects for some nematic equilibria on rectangular domains, Math. Mech. Solids, 25 (2020), pp. 1101-1123.
[12] D. Golovaty, J. A. Montero, and P. Sternberg, Dimension reduction for the Landau-de Gennes model in planar nematic thin films, J. Nonlinear Sci., 25 (2015), pp. 1431-1451. · Zbl 1339.35241
[13] Y. Han, Y. Hu, P. Zhang, and L. Zhang, Transition pathways between defect patterns in confined nematic liquid crystals, J. Comput. Phys., 396 (2019), pp. 1-11.
[14] Y. Han, Z. Xu, A.-C. Shi, and L. Zhang, Pathways connecting two opposed bilayers with a fusion pore: A molecularly-informed phase field approach, Soft Matter, 16 (2020), pp. 366-374.
[15] Y. Hu, Y. Qu, and P. Zhang, On the disclination lines of nematic liquid crystals, Commun. Comput. Phys., 19 (2016), pp. 354-379. · Zbl 1373.76012
[16] R. Ignat, L. Nguyen, V. Slastikov, and A. Zarnescu, Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 33 (2016), pp. 1131-1152. · Zbl 1351.82110
[17] M. Igor, S. Miha, T. Uros, R. Miha, and Z. Slobodan, Two-dimensional nematic colloidal crystals self-assembled by topological defects, Science, 313 (2006), pp. 954-958.
[18] S. Kralj and A. Majumdar, Order reconstruction patterns in nematic liquid crystal wells, Proc. A, 470 (2014), 20140276. · Zbl 1320.82067
[19] S. Kralj, R. Rosso, and E. G. Virga, Finite-size effects on order reconstruction around nematic defects, Phys. Rev. E(3), 81 (2010), 021702. · Zbl 1259.82120
[20] S. G. Krantz, Handbook of Complex Variables, Birkhäuser, Boston, 1999. · Zbl 0946.30001
[21] X. Lamy, Bifurcation analysis in a frustrated nematic cell, J. Nonlinear Sci., 24 (2014), pp. 1197-1230. · Zbl 1316.76009
[22] A. H. Lewis, I. Garlea, J. Alvarado, O. J. Dammone, P. D. Howell, A. Majumdar, B. M. Mulder, M. Lettinga, G. H. Koenderink, and D. G. Aarts, Colloidal liquid crystals in rectangular confinement: Theory and experiment, Soft Matter, 10 (2014), pp. 7865-7873.
[23] A. Logg, K.-A. Mardal, and G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lecture Notes Comput. Sci. Eng. 84, Springer, Berlin, 2012. · Zbl 1247.65105
[24] A. Majumdar, C. Newton, J. Robbins, and M. Zyskin, Topology and bistability in liquid crystal devices, Phys. Rev. E(3), 75 (2007), 051703.
[25] 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), pp. 227-280. · Zbl 1304.76007
[26] S. Mkaddem and E. Gartland Jr., Fine structure of defects in radial nematic droplets, Phys. Rev. E(3), 62 (2000), 6694.
[27] M. Muller, Y. G. Smirnova, G. Marelli, M. Fuhrmans, and A.-C. Shi, Transition path from two apposed membranes to a stalk obtained by a combination of particle simulations and string method, Phys. Rev. Lett., 108 (2012), 228103.
[28] I. Musevic and M. Skarabot, Self-assembly of nematic colloids, Soft Matter, 4 (2008), pp. 195-199.
[29] P. J. Olver, Complex Analysis and Conformal Mapping, University of Minnesota, Minneapolis, MN, 2017.
[30] P. Phillips and A. Rey, Texture formation mechanisms in faceted particles embedded in a nematic liquid crystal matrix, Soft Matter, 7 (2011), pp. 2052-2063.
[31] M. Robinson, C. Luo, P. E. Farrell, R. Erban, and A. Majumdar, From molecular to continuum modelling of bistable liquid crystal devices, Liquid Cryst., 44 (2017), pp. 2267-2284.
[32] A. Shams, X. Yao, J. O. Park, M. Srinivasarao, and A. D. Rey, Theory and modeling of nematic disclination branching under capillary confinement, Soft Matter, 8 (2012), pp. 11135-11143.
[33] K. Stephen and G. Adrian, Controllable alignment of nematic liquid crystals around microscopic posts: Stabilization of multiple states, Appl. Phys. Lett., 80 (2002), pp. 3635-3637.
[34] I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor & Francis, London, 2004.
[35] T. Takashi and N. Yasumasa, Morphological characterization of the diblock copolymer problem with topological computation, Jpn. J. Ind. Appl. Math., 27 (2010), pp. 175-190. · Zbl 1204.82044
[36] E. G. Virga, Variational Theories for Liquid Crystals, Appl. Math. 8, CRC Press, Boca Raton, FL, 1995. · Zbl 0814.49002
[37] Y. Wang, G. Canevari, and A. Majumdar, Order reconstruction for nematics on squares with isotropic inclusions: A Landau-de Gennes study, SIAM J. Appl. Math., 79 (2019), pp. 1314-1340. · Zbl 07098120
[38] H. H. Wensink, Polymeric nematics of associating rods: Phase behavior, chiral propagation, and elasticity, Macromolecules, 52 (2019), pp. 7994-8005.
[39] P. J. Wojtowicz, P. Sheng, and E. Priestley, Introduction to Liquid Crystals, Springer, Boston, 1975.
[40] X. Yao, H. Zhang, and J. Z. Chen, Topological defects in two-dimensional liquid crystals confined by a box, Phys. Rev. E(3), 97 (2018), 052707.
[41] J. Yin, Y. Wang, J. Z. Chen, P. Zhang, and L. Zhang, Construction of a pathway map on a complicated energy landscape, Phys. Rev. Lett., 124 (2020), 090601.
[42] J. Yin, L. Zhang, and P. Zhang, High-index optimization-based shrinking dimer method for finding high-index saddle points, SIAM J. Sci. Comput., 41 (2019), pp. A3576-A3595. · Zbl 1429.49030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.