Skip to main content
SpringerLink
Log in

Existence of minimizers for non-quasiconvex integrals

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J. J. Alibert & B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions, Arch. Rational Mech. Anal., 117 (1992), 155–166.

    Google Scholar 

  2. G. Aubert & R. Tahraoui, Théorèmes d'existence pour des problèmes du calcul des variations, J. Diff. Eqs., 33 (1979), 1–15.

    Google Scholar 

  3. G. Aubert & R. Tahraoui, Sur la minimisation d'une fonctionelle non convexe, non différentiable en dimension 1, Boll. Un. Mat. Ital., 17 (1980), 244–258.

    Google Scholar 

  4. G. Aubert & R. Tahraoui, Sur quelques résultats d'existence en optimisation non convexe, C.R. Acad. Sci. Paris, 297 (1983), 287–289.

    Google Scholar 

  5. G. Aubert & R. Tahraoui, Théorèmes d'existence en optimisation non convexe, Appl. Anal., 18 (1984), 75–100.

    Google Scholar 

  6. J. M. Ball & R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 15–52.

    Google Scholar 

  7. J. M. Ball & R. D. James, Proposed experimental tests of a theory of fine microstructure and the two well problem, Phil. Trans. Roy. Soc. London, A 338 (1991), 389–450.

    Google Scholar 

  8. K. Bhattacharya, N. B. Firoozye, R. D. James & R. V. Kohn, Restrictions on microstructure, Proc. Royal Soc. Edinburgh, 124 (1994), 843–878.

    Google Scholar 

  9. P. Bauman & D. Phillips, A nonconvex variational problem related to change of phase, J. Appl. Math. Optimization, 21 (1990), 113–138.

    Google Scholar 

  10. H. Brezis, Analyse fonctionnelle, théorié et applications, Masson, Paris. 1983.

    Google Scholar 

  11. G. Buttazzo, V. Ferone & B. Kawohl, Minimum problems over sets of concave functions and related questions, Math. Nachrichten, to appear.

  12. A. Cellina, On minima of a functional of the gradient: necessary conditions, Nonlinear Anal, Theory Meth. Appl, 20 (1993), 337–341.

    Google Scholar 

  13. A. Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal, Theory Meth. Appl, 20 (1993), 343–347.

    Google Scholar 

  14. A. Cellina & G. Colombo, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 7 (1990), 97–106.

    Google Scholar 

  15. A. Cellina & S. Zagatti, An existence result for a minimum problem in the vectorial case of the calculus of variations, S.I.A.M J. Control Optimization, to appear.

  16. A. Cellina & S. Zagatti, A version of Olech's lemma in a problem of the calculus of variations, S.I.A.M. J. Control Optimization, 32 (1994), 1114–1127.

    Google Scholar 

  17. L. Cesari, An existence theorem without convexity conditions, S.I.A.M. J. Control, 12 (1974), 319–331.

    Google Scholar 

  18. L. Cesari, Optimization — Theory and applications, Springer-Verlag, 1983.

  19. M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103 (1988), 237–277.

    Google Scholar 

  20. P. G. Ciarlet, Mathematical elasticity, Volume 1: Three-dimensional elasticity, North-Holland, 1988.

  21. P. G. Ciarlet & G. Geymonat, Sur les lois de comportement en élasticité non lineŕaire compressible, C. R. Acad. Sci. Paris, 295 (1982), 423–426.

    Google Scholar 

  22. A. Cutrì, Some remarks on Hamilton-Jacobi equations and non convex minimization problems, Rendiconti Mat., 13 (1993), 733–749.

    Google Scholar 

  23. B. Dacorogna, A relaxation theorem and its applications to the equilibrium of gases, Arch. Rational Mech. Anal., 77 (1981), 359–386.

    Google Scholar 

  24. B. Dacorogna, Direct methods in the calculus of variations, Springer-Verlag, 1989.

  25. I. Ekeland, Discontinuités de champs hamiltoniens et existence de solutions optimales en calcul des variations, Publ Math. (IHES), 47 (1977), 5–32.

    Google Scholar 

  26. I. Ekeland & R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, 1974.

    Google Scholar 

  27. N. B. Firoozye & R. V. Kohn, Geometric parameters and the relaxation of multiwell energies, in Microstructure and Phase Transition, edited by D. Kinderlehrer et al, Springer, 1992.

  28. I. Fonseca, Variational methods for elastic crystals, Arch. Rational Mech. Anal., 97 (1987), 189–220.

    Google Scholar 

  29. I. Fonseca & L. Tartar, The gradient theory of phase transition for systems with two potential wells, Proc. Royal Soc. Edinburgh, 111A (1989), 89–102.

    Google Scholar 

  30. R. L. Fosdick & G. Macsithigh, Helical shear of an elastic, circular tube with a nonconvex stored energy, Arch. Rational Mech. Anal., 84 (1983), 31–53.

    Google Scholar 

  31. G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Royal Soc. Edinburgh, 124 (1994), 437–471.

    Google Scholar 

  32. D. Giachetti & R. Schianchi, Minima of some non convex non coercive problems, Ann. Mat. Pura Appl., 165 (1993), 109–120.

    Google Scholar 

  33. M. E. Gurtin & R. Temam, On the anti-plane shear problem in finite elasticity, J. Elasticity, 11 (1981), 197–206.

    Google Scholar 

  34. Z. Hashin & S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials, J. Applied Phys., 33 (1962), 3125–3131.

    Google Scholar 

  35. G. Hellwig, Partial differential equations, Teubner, 1977.

  36. D. Kinderlehrer & P. Pedregal, Remarks about the analysis of gradient Young measures, Proc. Partial Differential Equations and Related Subjects, edited by M. Miranda, Pitman Research Notes in Math., 262 (1992), Longman, 125–150.

  37. R. Klötzler, On the existence of optimal processes, Banach Center Publications, Vol. 1, Warszawa, 1976, 125–130.

    Google Scholar 

  38. R. V. Kohn, The relaxation of a double-well energy, Continuum Mech. Thermodynamics, 3 (1991), 918–1000.

    Google Scholar 

  39. R. V. Kohn & G. Strang, Optimal design and relaxation of variational problems I, II, III, Comm. Pure Appl. Math., 39 (1986), 113–137, 139–182, 353–377.

    Google Scholar 

  40. H. Le Dret & A. Raoult, Enveloppe quasi-convexe de la densité d'énergie de Saint Venant-Kirchhoff, C. R. Acad. Sci Paris, 318 (1994), 93–98.

    Google Scholar 

  41. H. Le Dret & A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff energy function, preprint, 1994.

  42. P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman Research Notes in Math. 69, 1982.

  43. P. Marcellini, Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessita, Rendiconti di Matematica, 13 (1980), 271–281.

    Google Scholar 

  44. P. Marcellini, A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations, Mathematical Theories of Optimization, Proceedings, edited by J. P. Cecconi & T. Zolezzi, Springer Lecture Notes in Math., 979, 1983, 216–231.

  45. P. Marcellini, Some remarks on uniqueness in the calculus of variations, Collège de France Seminar, edited by H. Brezis & J.-L. Lions, Pitman Research Notes in Math., 84, 1983, 148–153.

  46. P. Marcellini, Non convex integrals of the calculus of variations, Methods of nonconvex Analysis, edited by A. Cellina, Springer Lecture Notes in Math., 1446, 1990, 16–57.

  47. P. Marcellini & C. Sbordone, Semicontinuity problems in the calculus of variations, Nonlinear Anal., Theory Meth. Appl., 4 (1980), 241–257.

    Google Scholar 

  48. E. Mascolo, Some remarks on non-convex problems, Material Instabilities in Continuum Mechanics, edited by J. M. Ball, Oxford Univ. Press, 1988, 269–286.

  49. E. Mascolo & R. Schianchi, Existence theorems for nonconvex problems, J. Math. Pures Appl., 62 (1983), 349–359.

    Google Scholar 

  50. E. Mascolo & R. Schianchi, Nonconvex problems of the calculus of variations, Nonlinear Anal., Theory. Meth. Appl., 9 (1985), 371–379.

    Google Scholar 

  51. E. Mascolo & R. Schianchi, Existence theorems in the calculus of variations, J. Diff. Eqs., 67 (1987), 185–198.

    Google Scholar 

  52. C. B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, 1966.

  53. S. Müller, Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations, Proceedings, Problems involving change of phase, Stuttgart, 1988, edited by K. Kirchgässner, Springer Lecture Notes in Physics, 1990, 31–44.

  54. C. Olech, Integrals of set valued functions and linear optimal control problems, Colloque sur la théorie mathématique du contrôle optimal, C.B.R.M. Louvain, 1970, 109–125.

    Google Scholar 

  55. A. C. Pipkin, Elastic materials with two prefered states, Quart. J. Mech. Appl. Math., 44 (1991), 1–15.

    Google Scholar 

  56. J. P. Raymond, Champs Hamiltoniens, relaxation et existence de solutions en calcul des variations, J. Diff. Eqs., 70 (1987), 226–274.

    Google Scholar 

  57. J. P. Raymond, Conditions nécessaires et suffisantes d'existence de solutions en calcul des variations, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 4 (1987), 169–202.

    Google Scholar 

  58. J. P. Raymond, Théorème d'existence pour des problèmes variationnels non convexes, Proc. Royal Soc. Edinburgh, 107A (1987), 43–64.

    Google Scholar 

  59. J. P. Raymond, Existence of minimizers for vector problems without quasiconvexity condition, Nonlinear Anal., Theory, Methods Appl., 18 (1992), 815–828.

    Google Scholar 

  60. P. Sternberg, Vector valued local minimizers of nonconvex variational problems, Rocky Mountain J. Math., 21 (1991), 799–807.

    Google Scholar 

  61. V. Sverak, On the problem of two wells, Geometric parameters and the relaxation of multiwell energies, in Microstructure and Phase Transition, edited by D. Kinderlehrer et al., Springer, 1992.

  62. R. Tahraoui, Théorèmes d'existence en calcul des variations et applications à l'élasticité non linéarie, C. R. Acad. Sci. Paris, 302 (1986), 495–498. Also in Proc. Royal Soc. Edinburgh, 109 A (1988), 51–78.

    Google Scholar 

  63. R. Tahraoui, Sur une classe de fonctionnelles non convexes et applications, S.I.A.M. J. Math. Anal., 21 (1990), 37–52.

    Google Scholar 

  64. L. Tartar, Estimations fines des coefficients homogénéisés, Ennio De Giorgi Colloquium, edited by P. Krée, Pitman Research Notes in Math. 125, 1985, 168–187.

  65. C. Truesdell & W. Noll, The nonlinear field theories of mechanics, in Handbuch der Physik, III/3, Springer-Verlag, 1965.

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne

    Bernard Dacorogna & Paolo Marcellini

  2. Dipartimento di Matematica, Viale Morgagni 67A, 50134, Firenze

    Bernard Dacorogna & Paolo Marcellini

Authors
  1. Bernard Dacorogna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Marcellini
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Brezis

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dacorogna, B., Marcellini, P. Existence of minimizers for non-quasiconvex integrals. Arch. Rational Mech. Anal. 131, 359–399 (1995). https://doi.org/10.1007/BF00380915

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00380915

Keywords

Search

Navigation