Function Spaces for Liquid Crystals

详细信息    查看全文

• 作者：Stephen Bedford
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：219
• 期：2
• 页码：937-984
• 全文大小：851 KB
• 参考文献：1.Alexander G.P., Yeomans J.M.: Stabilizing the blue phases. Phys. Rev. E 74(6), 061706 (2006)CrossRef ADS
  2.Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press Oxford, 2000
  3.Ambrosio, L., Tortorell, V.M.: On the approximation of free discontinuity problems. Scuola Normale Superiore (1990)
  4.Aviles, P., Giga, Y.: A mathematical problem related to the physical theory of liquid crystal configurations. Proc. Centre Math. Anal. Austral. Nat. Univ, vol. 12, pp. 1–16, 1987
  5.Ball J.M., Zarnescu A.: Orientability and energy minimization in liquid crystal models. Arch. Rat. Mech. Anal. 202(2), 493–535 (2011)MATH MathSciNet CrossRef
  6.Bedford, S.: Global minimisers of cholesteric liquid crystal systems (2014). arXiv:​1411.​3599
  7.Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. d’Analyse Mathématique 80(1), 37–86 (2000)MATH MathSciNet CrossRef
  8.Buttazzo, G., Michaille, G., Attouch, H.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, vol. 6. SIAM, Philadelphia, 2006
  9.Celada P.: Minimum problems on SBV with irregular boundary datum. Rend. Sem. Mat. Univ. Padova 98, 193–211 (1997)MATH MathSciNet
  10.Dávila J., Ignat R.: Lifting of BV functions with values in the unit circle. Comptes Rendus Mathematique 337(3), 159–164 (2003)MATH MathSciNet CrossRef
  11.Dym, H., McKean, H.P.: Fourier Series and Integrals, vol. 33. Academic Press, New York, 1972
  12.Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, vol. 5. CRC Press, New York, 1991
  13.Focardi M.: On the variational approximation of free-discontinuity problems in the vectorial case. Math. Models Meth. Appl. Sci. 11(04), 663–684 (2001)MATH MathSciNet CrossRef
  14.Frank F.C.: I. liquid crystals. on the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)CrossRef
  15.Kikuchi H., Yokota M., Hisakado Y., Yang H., Kajiyama T.: Polymer-stabilized liquid crystal blue phases. Nature Mater. 1(1), 64–68 (2002)CrossRef ADS
  16.Kleinert H., Maki K.: Lattice textures in cholesteric liquid crystals. Fortschritte der Physik 29(5), 219–259 (1981)MathSciNet CrossRef ADS
  17.Lavrentiev, M.: Sur quelques problémes du calcul des variations. An. Mat. Pura Appl., 7–28 (1926)
  18.Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics. American Mathematical Society, USA, 2009
  19.Lin, F-H.: A remark on the map x/[x]. Comptes Rendus de l’academie des Sciences Serie I-Mathematique, 305(12), 529–531 (1987)
  20.Majumdar A., Zarnescu A.: Landau–de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond. Arch. Rat. Mech. Anal. 196(1), 227–280 (2010)MATH MathSciNet CrossRef
  21.Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Mech. Anal. 98(2), 123–142 (1987)MATH MathSciNet CrossRef
  22.Nguyen L., Zarnescu A.: Refined approximation for minimizers of a Landau-de Gennes energy functional. Calculus Var. Part. Differ. Equ. 47(1–2), 383–432 (2013)MATH MathSciNet CrossRef
  23.Oseen C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29(140), 883–899 (1933)CrossRef MATH
  24.Oswald P., Baudry J., Pirkl S.: Static and dynamic properties of cholesteric fingers in electric field. Phys. Rep. 337(1), 67–96 (2000)CrossRef ADS
  25.Sethna J.P., Wright D.C., Mermin N.D.: Relieving cholesteric frustration: the blue phase in a curved space. Phys. Rev. Lett. 51(6), 467 (1983)MathSciNet CrossRef ADS
  26.Smalyukh I.I., Lansac Y., Clark N.A., Trivedi R.P.: Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. Nature Mater. 9(2), 139–145 (2010)CrossRef ADS
  27.Smalyukh, I.I., Senyuk, B.I., Palffy-Muhoray, P., Lavrentovich, O.D., Huang, H., Gartland Jr, E.C., Bodnar, V.H., Kosa, T., Taheri, B.: Electric-field-induced nematic-cholesteric transition and three-dimensional director structures in homeotropic cells. Phys. Rev. E 72(6), 061707 (2005)
  28.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions Elias M. Stein, vol. 2. Princeton University Press, Princeton, 1970
  29.Stewart, I.W.: The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. CRC Press, New York (2004)
  30.Virga, E.G.: Variational Theories for Liquid Crystals, vol. 8. CRC Press, New York, 1994
  31.Wojtowicz, P.J., Sheng, P., Priestley, E.B.: Introduction to Liquid Crystals. Springer, Berlin, 1975
  32.Wright D.C., Mermin N.D.: Crystalline liquids: the blue phases. Rev. Modern Phys. 61(2), 385 (1989)CrossRef ADS
  
• 作者单位：Stephen Bedford (1)
  
  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673

文摘

We consider the relationship between three continuum liquid crystal theories: Oseen–Frank, Ericksen and Landau–de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters s, n, and Q, we establish connections between the variational formulations of these theories. We use these results to justify a version of the Oseen–Frank theory using special functions of bounded variation. This proposed model can describe both orientable and non-orientable defects. Finally we study a number of frustrated nematic and cholesteric liquid crystal systems and show that the model predicts the existence of point and surface discontinuities in the director.