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Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings

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First European Congress of Mathematics Paris, July 6–10, 1992

Part of the book series: Progress in Mathematics ((PM,volume 120))

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Abstract

In the last decades there has been a growing interest in the general theory of variational problems for vector valued maps and in particular in geometric variational problems.

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References

  1. Almgrem F.J., Deformations and multiple-valued functions, in: Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. in Pure Math. 44, Amer. Math. Soc, Providence, 1986, 29–130.

    Google Scholar 

  2. Almgrem F.J., Browder W., Lieb E.H., Co-area, liquid crystals, and minimal surfaces, in: DDT — A Selection of Papers, Springer-Verlag, 1987.

    Google Scholar 

  3. Aviles P., Giga Y., Variational integrals on mappings of bounded variation and their lower semicontinuity, Arch. Rat. Mech. Anal. 115 (1991), 201–255.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal 63 (1977), 337–403.

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bethuel F., Brezis H., Coron J.M., Relaxed energies for harmonic maps, in: Variational methods, H. Berestycki, J.M. Coron, J. Ekeland (eds.), Birkhäuser, Basel, 1990.

    Google Scholar 

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

    Google Scholar 

  9. Brezis H., Coron J.M., Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), 203–215.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Eells J., Lemaire L., Report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68.

    Article  MathSciNet  MATH  Google Scholar 

  12. Eells J., Lemaire L., Selection topics in harmonic maps, CBMS 55, AMS, 1983.

    Google Scholar 

  13. Eells J., Lemaire L., Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524.

    Article  MathSciNet  MATH  Google Scholar 

  14. Ericksen J., Kinderlehrer D., Theory and applications of liquid crystals, IMA Series 5, Springer-Verlag, New York, 1969.

    Google Scholar 

  15. Federer H., Geometric Measure Theory, Springer-Verlag, New York, 1969.

    MATH  Google Scholar 

  16. Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Stud., Princeton University Press, 1983.

    MATH  Google Scholar 

  17. Giaquinta M., Modica G., Souček J., Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 106 (1989), 97–159

    Article  MATH  Google Scholar 

  18. Erratum and addendum, Arch. Rat. Mech. Anal. 109 (1990), 385–392.

    Article  MATH  Google Scholar 

  19. Giaquinta M., Modica G., Souček J., Cartesian currents and variational problems for mappings into spheres, Ann. Scuola Norm. Sup. Pisa 16 (1989), 393–485.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Giaquinta M., Modica G., Souček J., Partial regularity of cartesian currents which minimize certain variational integrals, in: PDE and calculus of variations, F. Colombini, A. Marino, L. Modica, S. Spagnolo (eds.), Birkhäuser, Basel, 1989.

    Chapter  Google Scholar 

  22. Giaquinta M., Modica G., Souček J., Liquid crystals: relaxed energies, dipoles, singular lines and singular points, Ann. Scuola Norm. Sup. Pisa 17 (1990), 415–437.

    MATH  Google Scholar 

  23. Giaquinta M., Modica G., Souček J., Cartesian currents and liquid crystals, singular lines and singular points, in Nematics, J.M. Coron, J.M. Ghidaglia, F. Hélein (eds.), Nato ASI Series 332, 1991.

    Google Scholar 

  24. Giaquinta M., Modica G., Souček J., The gap phenomenon for variational integrals in Sobolev spaces, Proc. Roy. Soc. Edinburgh 120 A (1992), 93–98.

    MATH  Google Scholar 

  25. Giaquinta M., Modica G., Souček J., The Dirichlet integral for mappings between manifolds: Cartesian Currents and Homology, Math. Ann. 294 (1992), 325–386.

    Article  MathSciNet  MATH  Google Scholar 

  26. Giaquinta M., Modica G., Souček J., Variational problems for the conformally invariant integral (math), Pitman Research Notes in Math. Series 267, 1992.

    Google Scholar 

  27. Giaquinta M., Modica G., Souček J., Variational problems for maps of bounded variation with values in S1, Calc. Var. 1 (1993), 87–121

    Article  MATH  Google Scholar 

  28. Giaquinta M., Modica G., Souček J., Graphs of finite mass which cannot be approximated in area by smooth graphs, Manuscripta Math. 78 (1993), 259–271.

    Article  MathSciNet  MATH  Google Scholar 

  29. Giaquinta M., Modica G., Souček J., Calculus of Variations and Cartesian Currents, in preparation.

    Google Scholar 

  30. Hardt R., Lin F.H., A remark on H 1 mappings, Manuscripta Math. 56 (1986), 1010.

    Article  MathSciNet  Google Scholar 

  31. Malý J., L p-approximation of Jacobians, Comment. Math. Univ. Carolinae 32(4) (1991), 659–666.

    MATH  Google Scholar 

  32. Sacks J., Uhlenbeck K., The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. Simon L., Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Canberra (1983).

    MATH  Google Scholar 

  35. White B., Infima of energy functionals in homotopy classes of mappings, J. Diff. Geom. 23 (1986), 127–142.

    MATH  Google Scholar 

  36. White B., Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

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© 1994 Birkhäuser Verlag

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Giaquinta, M. (1994). Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics Paris, July 6–10, 1992. Progress in Mathematics, vol 120. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9112-7_3

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  • DOI: https://doi.org/10.1007/978-3-0348-9112-7_3

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9912-3

  • Online ISBN: 978-3-0348-9112-7

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