Point and Line Singularities in Liquid Crystals

  • Robert M. Hardt
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

A liquid crystal is generally understood to be a mesomorphic state of matter which flows like a liquid and which exhibits some anisotropic behavior. See [E], [EK], [C], [DG]. The liquid crystal phase usually lies between a solid phase and an isotropic liquid phase with phase transition being induced by temperature change. A static model typically involves a kinematic variable n(x), called the director, which is a unit vector defined for x in a spatial region Ω.

Keywords

Liquid Crystal Nematic Liquid Crystal Line Singularity Partial Regularity Unique Continuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    F.J. Almgren, Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension 2, preprint.Google Scholar
  2. [AL]
    F.J. Almgren, Jr. and E. Lieb, Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds, Ann. Math., 128 (1988) 483–530.MathSciNetCrossRefMATHGoogle Scholar
  3. [BCL]
    H. Brezis, J.-M. Coron, and E. Lieb, Harmonic maps with defects, Comm. Math. Physics 107 (1986), 649–705.MathSciNetCrossRefMATHGoogle Scholar
  4. [C]
    S. Chandrasekhar, Liquid Crystals, Cambridge U. Press, 1977.Google Scholar
  5. [CHKLL]
    R. Cohen, R. Hardt, D. Kinderlehrer, S.-Y. Lin, and M. Luskin, Minimum energy configurations for liquid crystals: computational results in Theory and applications of liquid crystals, IMA vol. 5, Springer, 1986, 99–122.Google Scholar
  6. [D]
    C. Dafermos, Disclinations in liquid crystals, Quart. J. Mech. Appl. Math. 23.2 (1970), 49–64.Google Scholar
  7. [DG]
    P.G. DeGennes, The Physics of Liquid Crystals, Clarendon Press, 1974.Google Scholar
  8. [E]
    J. Ericksen, Equilbrium theory of liquid crystals, Adv. in liquid crystals 2, (G.H. Brown, ed.), Academic Press (1976), 233–298.Google Scholar
  9. [EK]
    J. Ericksen and D. Kinderlehrer, ed., Theory and Applications of Liquid Crystals, IMA vol. 5, Springer, 1986.Google Scholar
  10. [FF]
    F. Frank, Liquid crystals, Discuss. Faraday Soc. 25 (1958), 19–28.CrossRefGoogle Scholar
  11. [FG]
    G. Freidel, Annals Phys. 9e serie, 18 (1922), 273–474.Google Scholar
  12. [GL]
    N. Garofalo and F.H. Lin, Monotonicity properties of variational integrals, A p -weights, and unique continuation, Ind. U. Math. J. 35 (1986), 245–268.Google Scholar
  13. [GW]
    R. Gulliver and B. White, The rate of convergence of a harmonic map at a singular point, Math. Annalen 283 (1989), 539–550.MathSciNetCrossRefMATHGoogle Scholar
  14. [H]
    F. Hélein, Minima de la fonctionnelle énergie libre des cristaux liquides, to appear in Comp. Rend. A.S.Google Scholar
  15. [HKL1]
    R. Hardt, D. Kinderlehrer, and F.H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Physics 105 (1986), 547–570.MathSciNetCrossRefMATHGoogle Scholar
  16. [HKL2]
    R. Hardt, D. Kinderlehrer, and F.H. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincare, Anal. Nonlin. 5, no. 4 (1988), 297–322.MathSciNetMATHGoogle Scholar
  17. [HKW]
    S. Hildebrandt, H. Kaul, and K. Widman, An existence theory for harmonic mappings of Riemannian manifolds, Acta. Math. 138 (1977), 1–16.MathSciNetMATHGoogle Scholar
  18. [HL1]
    R. Hardt and F.H. Lin, A remark on H 1 mappings, Manuscripta Math. 56 (1986), 1–10.MathSciNetCrossRefMATHGoogle Scholar
  19. [HL2>]
    R. Hardt and F.H. Lin, Stability of singularities of minimizing harmonic maps, to appear in J. Diff. Geom.Google Scholar
  20. [HL3]
    R. Hardt and F.H. Lin, Line singularities in liquid crystals,in preparation.Google Scholar
  21. [L1]
    F.H. Lin, Une remarque sur l’application xl1x1, Comp. Rend. A.S. 305–1 (1987), 529–531.Google Scholar
  22. [L2]
    F.H. Lin, Nonlinear theory of defects in nematic liquid crystals—phase transition and flow phenomena, preprint.Google Scholar
  23. [M]
    M] J. Maddocks, A model for disclinations in nematic liquid crystals,IMA, vol. 5, Springer, 1986, 255–269.Google Scholar
  24. [RWB]
    C. Robinson, J.C. Ward, and R.B. Beevers, Discuss. Faraday Soc. 25 (1958), 29–42.CrossRefGoogle Scholar
  25. [S]
    L. Simon, Isolated singularities for extrema of geometric variational problems, Springer Lecture Notes 1161, 1985.Google Scholar
  26. [SU1]
    R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307–335.MathSciNetMATHGoogle Scholar
  27. [SU2]
    R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem of harmonic maps, J. Diff. Geom. 18 (1983), 253–268.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Robert M. Hardt
    • 1
  1. 1.Mathematics DepartmentRice UniversityHoustonUSA

Personalised recommendations