Communications in Mathematical Physics

, Volume 107, Issue 4, pp 649–705 | Cite as

Harmonic maps with defects

  • Haïm Brezis
  • Jean-Michel Coron
  • Elliott H. Lieb


Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation. Extensions of these problems are also solved, e.g. points are replaced by “holes,” ℝ3,S2 is replaced by ℝ N ,SN−1 or by ℝ N , ℝPN−1, the latter being appropriate for the theory of liquid crystals.


Neural Network Statistical Physic Complex System Liquid Crystal Nonlinear Dynamics 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, R.A.: Sobolev spaces. New York: Academic Press 1975Google Scholar
  2. 2.
    Birkhoff, G.: Tres observaciones sobre el algebra lineal. Univ. Nac. Tucuman Revista A.5, 147–151 (1946); Math. Rev.8, 561 (1947)Google Scholar
  3. 3.
    Brezis, H.: Analyse fonctionnelle. Paris: Masson 1983Google Scholar
  4. 4.
    Brezis, H., Coron, J.-M.: Large solutions for harmonic maps in two dimensions. Commun. Math. Phys.92, 203–215 (1983)Google Scholar
  5. 5.
    Brezis, H., Coron, J.-M., Lieb, E.H.: Estimations d'énergie pour des applications de ℝ3 à valeurs dansS 2. C.R. Acad. Sci. Paris303, 207–210 (1986)Google Scholar
  6. 6.
    Brinkman, W.F., Cladis, P.E.: Defects in liquid crystals. Phys. Today, May 1982, pp. 48–54Google Scholar
  7. 7.
    Chandrasekhar, S.: Liquid crystals. Cambridge: Cambridge University Press 1977Google Scholar
  8. 8.
    Cohen, R., Hardt, R., Kinderlehrer, D., Lin, S.-Y., Luskin, M.: Minimum energy configurations for liquid crystals: Computational results, to appear in ref. [14]Google Scholar
  9. 9.
    De Gennes, P.G.: The physics of liquid crystals. Oxford: Clarendon Press 1974Google Scholar
  10. 10.
    Dudley, R.M.: Probabilities and metrics. Aarhus Universitet, Matematisk Institut Lecture Notes Series no 45 (1976)Google Scholar
  11. 11.
    Dunford, N., Schwartz, J.T.: Linear operators, Vol. I. New York: Interscience 1964Google Scholar
  12. 12.
    Ekeland, I., Temam, R.: Analyse convexe et problèmes variationnels. Paris: Dunod, Gauthier-Villars 1974Google Scholar
  13. 13.
    Ericksen, J.L.: Equilibrium theory of liquid crystals. In: Advances in liquid crystals, Vol. 2. Brown, G.H. (ed.). New York: Academic Press 1976, pp. 233–299Google Scholar
  14. 14.
    Ericksen, J.L., Kinderlehrer, D. (ed.): Proceedings I.M.A. workshop on the theory and applications of liquid crystals (to appear)Google Scholar
  15. 15.
    Fernique, X.: Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais. In: Sem. Probabilités XV. Lecture Notes in Mathematics, No. 850, Azema-Yor (ed.). Berlin, Heidelberg, New York: Springer 1981Google Scholar
  16. 16.
    Hamidoune, Y.O., Las Vergnas, M.: Local edge-connectivity in regular bipartite graphs (to appear)Google Scholar
  17. 17.
    Hardt, R., Kinderlehrer, D.: Mathematical questions of liquid crystal theory. To appear in ref. [14]Google Scholar
  18. 18.
    Hardt, R., Kinderlehrer, D., Lin, F.H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys.105, 541–570 (1986)Google Scholar
  19. 19.
    Hardt, R., Kinderlehrer, D., Lin, F.H.: In preparationGoogle Scholar
  20. 20.
    Kantorovich, L.V.: On the transfer of masses. Dokl. Akad. Nauk SSSR37, 227–229 (1942)Google Scholar
  21. 21.
    Kléman, M.: Points, lignes, parois, Vol. I and II. Orsay: Les editions de physique 1977Google Scholar
  22. 22.
    Lemaire, L.: Applications harmoniques de surfaces Riemanniennes. J. Differ. Geom.13, 51–78 (1978)Google Scholar
  23. 23.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. Riv. Mat. Iberoamericana1, 45–121 and 145–201 (1985)Google Scholar
  24. 24.
    Lloyd, N.G.: Degree theory. Cambridge: Cambridge University Press 1978Google Scholar
  25. 25.
    Meyers, N., Serrin, J.:H=W. Proc. Nat. Acad. Sci. USA51, 1055–1056 (1964)Google Scholar
  26. 26.
    Minc, H.: Permanents, Encyclopedia of Math. and Appl., Vol. 6. Reading, MA: Addison-Wesley 1978Google Scholar
  27. 27.
    Nirenberg, L.: Topics in nonlinear functional analysis. New York: New York University Lecture Notes 1974Google Scholar
  28. 28.
    Parisi, G.: Quark imprisonment and vacuum repulsion. Phys. Rev. D11, 970–971 (1975)Google Scholar
  29. 29.
    Rachev, S.T.: The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl.29, 647–676 (1985)Google Scholar
  30. 30.
    Rockafellar, R.T.: Extension of Fenchel's duality theorem for convex functions. Duke Math. J.33, 81–90 (1966)Google Scholar
  31. 31.
    Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom.17, 307–335 (1982)Google Scholar
  32. 32.
    Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom.18, 253–268 (1983)Google Scholar
  33. 33.
    Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math.118, 525–571 (1983)Google Scholar
  34. 34.
    Spanier, E.H.: Algebraic topology. New York: McGraw-Hill 1966Google Scholar
  35. 35.
    Springer, G.: Introduction to Riemann surfaces. Reading, MA: Addison-Wesley 1957Google Scholar
  36. 36.
    Strang, G.:L 1 andL approximation of vector fields in the plane. In: Lecture Notes in Num. Appl. Anal.5, 273–288 (1982)Google Scholar
  37. 37.
    Whitney, H.: Geometric integration theory. Princeton, NJ: Princeton University Press 1957Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Haïm Brezis
    • 1
  • Jean-Michel Coron
    • 2
  • Elliott H. Lieb
    • 3
  1. 1.Département de MathématiquesUniversité P. et M. CurieParis Cedex 05France
  2. 2.Département de MathématiquesEcole PolytechniquePalaiseau CedexFrance
  3. 3.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

Personalised recommendations