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New energies for harmonic maps and liquid crystals

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References

  1. F. Almgren-E. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples and bounds, Annals of Math. 129 (1988), 483–530.

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Bethuel, A characterization of maps in H 1 (B 3, S 2) which can be approximated by smooth maps, Ann. IHP Analyse Nonlinéaire 7 (1990), 269–286.

    MathSciNet  MATH  Google Scholar 

  3. F. Bethuel, The approximation problem for Sobolev maps between manifolds, Acta Math. 167 (1991), 153–206.

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Bethuel-H. Brezis, Regularity of minimizers of relaxed problems for harmonic maps, J. Funct. Anal. 101 (1991), 145–161.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Bethuel-H. Brezis-J.M. Coron, Relaxed energies for harmonic maps, in Variational Problems (H. Berestycki, J.M. Coron and I. Ekeland ed.), Birkhauser (1990).

    Google Scholar 

  6. F. Bethuel-X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988), 60–75.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Brezis, Liquid crystals and energy estimates for S 2-valued maps, in Theory and Applications of Liquid Crystals (J. Ericksen and D. Kinderlehrer ed.), Springer (1987).

    Google Scholar 

  8. H. Brezis, S k-valued maps with singularities, in Topics in the Calculus of Variations (M. Giaquinta ed.), Lecture Notes in Math. vol. 1365, Springer (1989), 1–30.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  11. P. DeGennes, The Physics of Liquid Crystals, Clarendon Press, Oxford (1974).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Ericksen-D. Kinderlehrer ed., Theory and Applications of Liquid Crystals, IMA Series vol. 5, Springer (1987).

    Google Scholar 

  14. C. Evans, Partial regularity for stationary harmonic maps into the sphere, Archive Rat. Mech. Anal. 116 (1991), 101–113.

    Article  MATH  Google Scholar 

  15. M. Giaquinta-G. Modica-J. Soucek, Cartesian currents and variational problems for mappings into spheres, Ann. Sc. Norm. Sup. Pisa 16 (1989), 393–485.

    MathSciNet  MATH  Google Scholar 

  16. M. Giaquinta-G. Modica-J. Soucek, The Dirichlet energy of mappings with values into the sphere, Manuscripta Math. 65 (1989), 489–507.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Hajlasz, Approximation of Sobolev mapping, Diff. and Int. Eq. (to appear).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Hardt-F.H. Lin-C.C. Poon, Axially symmetric harmonic maps minimizing a relaxed energy (to appear).

    Google Scholar 

  20. D. Kinderlehrer, Recent developments in liquid crystal theory, Proc. Conf. in honor of J.L. Lions (R. Dautray ed.), North Holland (1991).

    Google Scholar 

  21. T. Rivière, Applications harmoniques de B 3 dans S 2 ayant une ligne de singularités, CRAS 313 (1991), 583–587.

    MATH  Google Scholar 

  22. T. Rivière, Applications harmoniques de B 3 dans S 2 partout discontinues, CRAS 314 (1992), 719–723 and detailed paper to appear.

    MathSciNet  Google Scholar 

  23. R. Schoen-K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307–335 and Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983), 253–268.

    MathSciNet  MATH  Google Scholar 

  24. J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139–167.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Yosida, Functional Analysis, Springer (1965).

    Google Scholar 

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Hikosaburo Komatsu

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Dedicated to the memory of Professor K. Yosida

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© 1993 Springer-Verlag

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Brezis, H. (1993). New energies for harmonic maps and liquid crystals. In: Komatsu, H. (eds) Functional Analysis and Related Topics, 1991. Lecture Notes in Mathematics, vol 1540. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085471

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  • DOI: https://doi.org/10.1007/BFb0085471

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  • Print ISBN: 978-3-540-56471-3

  • Online ISBN: 978-3-540-47565-1

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