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Sk—Valued maps with singularities

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1365)

Keywords

  • Liquid Crystal
  • Boundary Data
  • Geodesic Distance
  • Degree Zero
  • Jacobian Determinant

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. F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. 27 (1968), P. 321–391.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. F. Almgren — W. Browder — E. Lieb, Co-area, liquid crystals, and minimal surfaces in DD7 — a selection of papers, Springer (1987).

    Google Scholar 

  3. F. Almgren-E. Lieb, Singularities of energy minimizing maps from the ball to the sphere, Bull. Amer. Math. Soc. 17 (1987), p. 304–306 and detailed paper to appear.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. F. Bethuel, Approximation dans des espaces de Sobolev entre deux variétés et groupes d'homotopie, C.R. Acad. Sc. Paris (1988) and detailed paper to appear.

    Google Scholar 

  5. F. Bethuel, A characterization of maps in H1(B3,S2), which can be approximated by smooth maps, to appear.

    Google Scholar 

  6. F. Bethuel-X. Zheng, Sur la densité des fonctions régulières entre deux variétés dans les espaces de Sobolev, C.R. Acad. Sc. Paris 303 (1986), p. 447–447 and Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. (to appear).

    MathSciNet  MATH  Google Scholar 

  7. H. Brézis, Liquid crystals and energy estimates for S2-valued maps, in [16].

    Google Scholar 

  8. H. Brézis-J. M. Coron-E. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), p. 649–705.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. W. Brinkman-P. Cladis, Defects in liquid crystals, Physics Today, May 1982, p. 48–54.

    Google Scholar 

  10. S. Chandrasekhar, Liquid crystals, Cambridge University Press (1977).

    Google Scholar 

  11. R. Cohen-R. Hardt-D. Kinderlehrer-S. Y. Lin-M. Luskin, Minimum energy configurations for liquid crystals: computational results in [16].

    Google Scholar 

  12. J. M. Coron-R. Gulliver, Minimizing p-Harmonic maps into spheres (to appear).

    Google Scholar 

  13. P. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford (1974).

    MATH  Google Scholar 

  14. E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Boll U.M.I. 4 (1968), p. 135–137.

    MATH  Google Scholar 

  15. J. Eells-L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), p 1–68.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. Ericksen, Equilibrium theory of liquid crystals, in Advances in Liquid Crystals 2, (G.Brown ed.), Acad. Press (1976), p. 233–299.

    Google Scholar 

  17. J. Ericksen, Static theory of point defects in nematic liquid crystals (to appear).

    Google Scholar 

  18. J. Ericksen-D. Kinderlehrer ed., Theory and applications of liquid crystals, IMA Series Vol. 5, Springer (1987).

    Google Scholar 

  19. M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of SM-valued functions (to appear).

    Google Scholar 

  20. H. Federer, Geometric measure theory, Springer (1969).

    Google Scholar 

  21. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princton Univ. Press (1983).

    Google Scholar 

  22. E. Guisti-M. Miranda, Sulla regularita delle soluzioni deboli di una classe di sistemi ellitici quasilineari, Arch. Rat. Mech. Anal. 31 (1968), p. 173–184.

    Google Scholar 

  23. Y. Hamidoune-M. Las Vergnas, Local edge-connecting in regular bipartite graphs (to appear).

    Google Scholar 

  24. R. Hardt, An introduction to geometric measure theory, Lecture Notes, Melbourne Univ. (1979).

    Google Scholar 

  25. R. Hart-D. Kinderlehrer-F. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), p. 547–570.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. R. Hardt-D. Kinderlehrer-F. Lin, Stable defects of minimizers of constrained variational principles, Ann. I.H.P., Analyse Nonlinéaire (to appear).

    Google Scholar 

  27. R. Hardt-F. Lin, A remark on H1 mappings, Manuscripta Math. 56 (1986), p. 1–10.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. R. Hardt-F. Lin, Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math. 40 (1987), p. 556–588.

    CrossRef  MathSciNet  Google Scholar 

  29. R. Hardt-F. Lin, Stability of singularities of minimizing harmonic maps, Math. Annalen (to appear).

    Google Scholar 

  30. F. Hélein, Minima de la fonctionnelle energie libre des cristaux liquides, C.R. Acad. Sc. Paris 305 (1987), p. 565–568.

    MathSciNet  MATH  Google Scholar 

  31. W. Jager-H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math. 343 (1983), p. 146–161.

    MathSciNet  MATH  Google Scholar 

  32. L. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk SSSR 37 (1942), p. 227–229.

    Google Scholar 

  33. M. Kleman, Points, lines and walls, John-Wiley (1983).

    Google Scholar 

  34. F. Lin, Une remarque sur l'application x/|x|, C.R. Acad. Sc. Paris (à paraître).

    Google Scholar 

  35. C. Morrey, Partial regularity results for nonlinear elliptic systems, J. Math. Mech. 17 (1968), p. 649–670.

    MathSciNet  MATH  Google Scholar 

  36. L. Nirenberg, Topics in Nonlinear Functional Analysis, N.Y.U. Lecture Notes, New York (1974).

    Google Scholar 

  37. S. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications, Theory of Prob. and Applic. 29 (1985), p. 647–676.

    CrossRef  MATH  Google Scholar 

  38. R. Schoen-K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), p. 307–335.

    MathSciNet  MATH  Google Scholar 

  39. R. Schoen-K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983), p. 253–268.

    MathSciNet  MATH  Google Scholar 

  40. R. Schoen-K. Uhlenbeck, Approximation theorems for Sobolev mappings (to appear).

    Google Scholar 

  41. L. Simon, Lectures on geometric measure theory, Canberra (1984).

    Google Scholar 

  42. L. Simon, Asymptotics for a class of nonlinear evolution equations with applications to geometric problems, Ann. of Math. 118 (1983), p. 525–571.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. B. White, Infina of energy functionals in homotopy classes, J. Diff. Geom. 23 (1986), p. 127–142.

    MATH  Google Scholar 

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© 1989 Springer-Verlag

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Brézis, H. (1989). Sk—Valued maps with singularities. In: Giaquinta, M. (eds) Topics in Calculus of Variations. Lecture Notes in Mathematics, vol 1365. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089176

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  • DOI: https://doi.org/10.1007/BFb0089176

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