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The Variety of Configurations of Static Liquid Crystals

  • Robert Hardt
  • David Kinderlehrer
  • Fang Hau Lin
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

Here we make several observations on a variety of special classes of harmonic maps from domains in IR3 to S 2. Such maps are relevant for the study of liquid crystals; see e.g. [HK1], [HKL1], [HKLu]. Part of our work is motivated by questions about the nonuniqueness and the number of harmonic maps having fixed boundary data. Here we show that the number may actually be infinite. For example, by Corollary 3.2 below there exists a one parameter family of distinct energy-minimizers each having the same Dirichlet boundary data.

Keywords

Liquid Crystal Singular Point Boundary Data Minimal Hypersurface Monotonicity Formula 
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.

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References

  1. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [B]
    A. Baldes, Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Z. 185 (1984), 505–516.MathSciNetCrossRefGoogle Scholar
  3. [BCL]
    H. Brezis, J.-M. Coron, and E. Lieb, Harmonic maps with defects, Comm. Math. Physics 107 (1986), 649–705.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [C]
    R. Courant, Dirichlet’s principle, Interscience, 1950.Google Scholar
  5. [CH]
    J.-M. Coron and F. Helein, Harmonic diffeomorphisms, minimizing harmonic maps, and rotational symmetry, preprint, Univ. Paris Sud, Orsay, 1988.Google Scholar
  6. [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, 5, Springer, 1986.Google Scholar
  7. [D]
    Ding, Symmetric harmonic maps between spheres, preprint.Google Scholar
  8. [E]
    J. Ericksen, Static theory of point defects in nematic liquid crystals, preprint.Google Scholar
  9. [GW]
    R. Gulliver and B. White, The rate of convergence of a harmonic map at a singular point, Math. Ann. 283 (1989), 539–550.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Ha]
    R. Hardt, Point and line singularities in liquid crystals, this volume.Google Scholar
  11. [He]
    F. Helein, Minima de la fonctionelle éenergie libre des cristaux liquides, Calcul des Variations, to appear in C.R.A.S.Google Scholar
  12. [HK1]
    R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystal theory, in Theory and Applications of Liquid Crystals, IMA, 5, Springer, 1986.Google Scholar
  13. [HK2]
    R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystal theory, in College de France Seminar, vol. IV(3), 1987.Google Scholar
  14. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [HKL2]
    R. Hardt, D. Kinderlehrer, and F.H. Lin, Stable defects of minimizers of constrained variational problems, Ann. Inst. Henri Poincare, Analyse non lineaire, 5, no. 4 (1988), 297–322.MathSciNetzbMATHGoogle Scholar
  16. [HKLu]
    R. Hardt, D. Kinderlehrer, and M. Luskin, Remarks about the mathematical theory of liquid crystals, in Calculus of Variations and Partial Differential Equations, S. Hildebrandt, D. Kinderlehrer, and M. Miranda, ed., Lecture Notes in Math. 1340, 123–138.Google Scholar
  17. [HKW]
    S. Hildebrandt, H. Kaul, and K. Widman, An existence theory for harmonic maps of Riemannian manifolds, Acta Math. 188 (1977), 116.MathSciNetGoogle Scholar
  18. [HL1]
    R. Hardt and F.H. Lin, A remark on H 1 mappings, Manuscripta Math. 56 (1986), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [HL2]
    R. Hardt and F.H. Lin, Mappings minimizing the LP norm of the gradient, Comm. Pure & Appl. Math. 40 (1987), 555–588.MathSciNetzbMATHGoogle Scholar
  20. [HL3]
    R. Hardt, and F.H. Lin, Stability of singularities of minimizing harmonic maps, to appear in J. Dif£. Geom. 28 (1988).Google Scholar
  21. [JK1]
    W. Jager and H. Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, Manuscripta Math. 28 (4) (1979), 269–271.MathSciNetCrossRefGoogle Scholar
  22. [JK2]
    W. Jager and H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions to an elliptic systems, J. Reine Angew. Math. 343 (1983), 146–161.MathSciNetGoogle Scholar
  23. [K]
    D. Kinderlehrer, Recent developments in liquid crystal theory,Proceedings of Colloq. Lions.Google Scholar
  24. [Ka]
    B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math. 1150, Springer-Verlag, Berlin-Heidelberg-New York, 1985.Google Scholar
  25. [La]
    H.B. Lawson, The equivariant Plateau problem, Trans. A.M.S. 173 (1972), 231–250.Google Scholar
  26. [LSY]
    Lin, San-Yih, Numerical analysis of liquid crystal problems, Thesis, U. of Minnesota, 1987.Google Scholar
  27. [LFH]
    Lin, Fang-hua, Nonlinear theory of defects in nematic liquid crystals —phase transition and flow phenomena,to appear in Comm. Pures & Appl. Math.Google Scholar
  28. [LM]
    J.L. Lions and E. Magenes, Nonhomogeneous boundary value problemsand applications I, Springer-Verlag, Berlin-Heidelberg-New York, 1962.Google Scholar
  29. [M1]
    F. Morgan, A smooth curve in R 3 bounding a continuum of minimal surfaces, Arch. Rat. Mech. Anal. 75 (1981), 193–197.Google Scholar
  30. [M2]
    F. Morgan, Finiteness of the number of stable minimal hypersurfaces with a fixed boundary, Indiana U. Math. J. 35 (4) (1986), 779–833.MathSciNetGoogle Scholar
  31. [S]
    L. Simon, Isolated singularities of extrema of geometric variational problems, Lecture Notes in Math. 1186, Springer-Verlag, Berlin-Heidelberg -New York, 1986.Google Scholar
  32. [SaU]
    J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1–24.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Sm]
    R.T. Smith, Harmonic maps of spheres, Amer. J. Math. 97 (2) (1975), 364–385.CrossRefzbMATHGoogle Scholar
  34. [Sma]
    N. Smale, Minimal hypersurfaces with many isolated singularities, Ann. Math. 130 (1989), 603–642.CrossRefGoogle Scholar
  35. [St]
    E. Stein, Singular integrals and differentiability properties of functions, Princeton, 1970.Google Scholar
  36. [SU1]
    R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307–335.MathSciNetzbMATHGoogle Scholar
  37. [SU2]
    R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem of harmonic maps, J. Diff. Geom. 18 (1983), 253–268.MathSciNetzbMATHGoogle Scholar
  38. [SU3]
    R. Schoen and K. Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Inventiones Math. 78 (1984), 89–100.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Z]
    D. Zhang, Axially symmetric harmonic maps, preprint, Univ. Cal. San Diego, March, 1987.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Robert Hardt
    • 1
  • David Kinderlehrer
    • 2
  • Fang Hau Lin
    • 3
  1. 1.Mathematics DepartmentRice UniversityHoustonUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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