Skip to main content
Log in

Relaxation of some functionals of the calculus of variations

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

References

  1. J. J. Alibert andB. Dacorogna, An example of a quasiconvex function not polyconvex in dimension two. Arch. Rational Mech. Anal.117, 155–166 (1992).

    Google Scholar 

  2. J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.64, 337–403 (1977).

    Google Scholar 

  3. J. M. Ball andF. Murat,W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58, 225–253 (1984).

    Google Scholar 

  4. M.Bousselsal, Etude de quelques problémes de calcul des variations liés á la mécanique. Thesis, Université de Metz 1993.

  5. M. Chipot, Hyperelasticity for crystal. European J. App. Math.1, 113–129 (1990).

    Google Scholar 

  6. P. G.Ciarlet, Elasticité tridimensionnelle. Masson 1986.

  7. B.Dacorogna, Direct methods in the calculus of variations. Berlin-Heidelberg-New York 1989.

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

    Google Scholar 

  9. B. Dacorogna, Quasiconvexity and relaxation of non convex variational problem. J. Funct. Anal.46, 102–118 (1982).

    Google Scholar 

  10. B.Dacorogna, Weak continuity and weak lower semicontinuity of non linear functionals. LNM922, Berlin-Heidelberg-New York 1982.

  11. B. Dacorogna, Remarques sur les notions de polyconvexité, quasiconvexité et convexité de rang 1. J. Math. Pures Appl.64, 403–438 (1985).

    Google Scholar 

  12. B. Dacorogna, J. Douchet, W. Gangbo andJ. Rappaz, Some examples for rank one functions in dimension two. Proc. Roy. Soc. Edinburgh Sect. A114, 135–150 (1990).

    Google Scholar 

  13. N.Firoozye, Optimal translations and relaxations of some multiwell energies. Ph. D. Thesis, New York University 1990.

  14. W. Gangbo, On the continuity of the polyconvex, quasiconvex and rank one convex envelope with respect to growth condition. Proc. Roy. Soc. Edinburgh Sect. A123, 707–729 (1993).

    Google Scholar 

  15. R. V.Kohn, The relaxation of a double-well energy. To appear.

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

    Google Scholar 

  17. H. LeDrei and A.Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff Energy. To appear.

  18. C. B. Morrey, Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math.2, 25–53 (1952).

    Google Scholar 

  19. C. B.Morrey, Multiple integrals in the calculus of variations. Berlin-Heidelberg-New York 1966.

  20. V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A120, 185–189 (1992).

    Google Scholar 

  21. V. Šverák, Examples of rank one convex functions. Proc. Roy. Soc. Edinburgh Sect. A114, 237–242 (1990).

    Google Scholar 

  22. V. Šverák, Quasiconvexity functions with subquadratic growth. Proc. Roy. Soc. London Sect. A433, 723–725 (1991).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bousselsal, M., Chipot, M. Relaxation of some functionals of the calculus of variations. Arch. Math 65, 316–326 (1995). https://doi.org/10.1007/BF01195543

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01195543