manuscripta mathematica

, Volume 51, Issue 1–3, pp 1–28 | Cite as

Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals

  • Paolo Marcellini
Article

Abstract

We study semicontinuity of multiple integrals ∫Ωf(x,u,Du) dx, where the vector-valued function u is defined for\(x \varepsilon \Omega \subset \mathbb{R}^n \) with values in ℝN. The function f(x,s,ξ) is assumed to be Carathéodory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology of the Sobolev space H1,p(ΩℝN). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them beingconvex andindependent of (x,s) for large values of ξ. In the special polyconvex case, for example if n=N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of H1,p(Ωℝn) for small p, in particular for some p smaller than n.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    ACERBI E., BUTTAZZO G., FUSCO N., Semicontinuity and relaxation for integral depending on vector-valued functions, J. Math. Pures Appl., 62 (1983), 371–387Google Scholar
  2. [2]
    ACERBI E., FUSCO N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., to appearGoogle Scholar
  3. [3]
    ANTMAN S.S., The influence of elasticity on analysis: modern developments, Bull. Amer. Math. Soc., 9 (1983), 267–291Google Scholar
  4. [4]
    ATTOUCH H., SBORDONE C., Asymptotic limits for perturbed functionals of calculus of variations, Ricerche Mat., 29 (1980), 85–124Google Scholar
  5. [5]
    BALL J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337–403Google Scholar
  6. [6]
    BALL J.M., CURRIE J.C., OLVER P.J., Null Lagragians, weak continuity and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135–175Google Scholar
  7. [7]
    DACOROGNA B., Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal., 46 (1982), 102–118Google Scholar
  8. [8]
    DACOROGNA B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Math., 922 (1982), Springer-Verlag, BerlinGoogle Scholar
  9. [9]
    DAL MASO G., MODICA L., A general theory of variational functionals, Topics in Func. Anal. 1980–81, Quaderno Scuola Norm. Sup. Pisa, 1981, 149–221Google Scholar
  10. [10]
    DE GIORGI E., Teoremi di semicontinuità nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968–69Google Scholar
  11. [11]
    DE GIORGI E., Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rendiconti Mat., 8 (1975), 277–294Google Scholar
  12. [12]
    EISEN G., A counterexample for some lower semicontinuity results, Math. Z., 162 (1978), 241–243Google Scholar
  13. [13]
    EISEN G., A selection lemma for sequence of measurable sets, and lower semicontinuity of multiple integrals, Manuscripta Math., 27 (1979), 73–79Google Scholar
  14. [14]
    EKELAND I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443–474Google Scholar
  15. [15]
    EKELAND I., TEMAM R., Convex analysis and variational problems, North Holland, 1976Google Scholar
  16. [16]
    FUSCO N., Quasi convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat., 29 (1980), 307–323Google Scholar
  17. [17]
    GIAQUINTA M., GIUSTI E., On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31–46Google Scholar
  18. [18]
    GIAQUINTA M., GIUSTI E., Quasi-minima, 1983, preprintGoogle Scholar
  19. [19]
    MARCELLINI P., Some problems of semicontinuity and of Γ -convergence for integrals of the calculus of variations, Proc. Intern. Meet. on Recent Meth. in Nonlinear Anal., De Giorgi, Magenes, Mosco Edit., Pitagora Bologna, 1978, 205–221.Google Scholar
  20. [20]
    MARCELLINI P., Quasiconvex quadratic forms in two dimensions, Appl. Math. Optimization, 11 (1984), 183–189Google Scholar
  21. [21]
    MARCELLINI P., SBORDONE C., Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4 (1980), 241–257Google Scholar
  22. [22]
    MARCELLINI P., SBORDONE C., On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl., 62 (1983), 1–9Google Scholar
  23. [23]
    MEYERS N., An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189–206Google Scholar
  24. [24]
    MEYERS N., Quasiconvexity and lower semicontinuity of multiple integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125–149Google Scholar
  25. [25]
    MORREY C.B., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25–53Google Scholar
  26. [26]
    MORREY C.B., Multiple integrals in the calculus of variations, 1966, Springer-Verlag, BerlinGoogle Scholar
  27. [27]
    MURAT F., Compacité par compensation II, Proc. Inter. Meet. on Recent Meth. in Nonlinear Anal., De Giorgi, Magenes, Mosco Edit., Pitagora Bologna, 1978, 245–256Google Scholar
  28. [28]
    RESHETNYAK Y.G., General theorems on semicontinuity and on convergence with a functional, Sibirskii Math. J., 8 (1967), 1051–1069Google Scholar
  29. [29]
    RESHETNYAK Y.G., Stability theorems for mappings with bounded excursion, Sibirskii Math. J., 9 (1968), 667–684Google Scholar
  30. [30]
    SBORDONE C., Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa, 2 (1975), 617–638Google Scholar
  31. [31]
    SERRIN J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139–167Google Scholar
  32. [32]
    BALL J. M., MURAT F., W1,p-quasiconvexity and variational problems for multiple integrals, to appearGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Paolo Marcellini
    • 1
  1. 1.Dipartimento di Matematica IIUniversità di RomaRomaItaly

Personalised recommendations