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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ACERBI E., BUTTAZZO G., FUSCO N., Semicontinuity and relaxation for integral depending on vector-valued functions, J. Math. Pures Appl., 62 (1983), 371–387
ACERBI E., FUSCO N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., to appear
ANTMAN S.S., The influence of elasticity on analysis: modern developments, Bull. Amer. Math. Soc., 9 (1983), 267–291
ATTOUCH H., SBORDONE C., Asymptotic limits for perturbed functionals of calculus of variations, Ricerche Mat., 29 (1980), 85–124
BALL J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337–403
BALL J.M., CURRIE J.C., OLVER P.J., Null Lagragians, weak continuity and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135–175
DACOROGNA B., Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal., 46 (1982), 102–118
DACOROGNA B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Math., 922 (1982), Springer-Verlag, Berlin
DAL MASO G., MODICA L., A general theory of variational functionals, Topics in Func. Anal. 1980–81, Quaderno Scuola Norm. Sup. Pisa, 1981, 149–221
DE GIORGI E., Teoremi di semicontinuità nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968–69
DE GIORGI E., Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rendiconti Mat., 8 (1975), 277–294
EISEN G., A counterexample for some lower semicontinuity results, Math. Z., 162 (1978), 241–243
EISEN G., A selection lemma for sequence of measurable sets, and lower semicontinuity of multiple integrals, Manuscripta Math., 27 (1979), 73–79
EKELAND I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443–474
EKELAND I., TEMAM R., Convex analysis and variational problems, North Holland, 1976
FUSCO N., Quasi convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat., 29 (1980), 307–323
GIAQUINTA M., GIUSTI E., On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31–46
GIAQUINTA M., GIUSTI E., Quasi-minima, 1983, preprint
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.
MARCELLINI P., Quasiconvex quadratic forms in two dimensions, Appl. Math. Optimization, 11 (1984), 183–189
MARCELLINI P., SBORDONE C., Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4 (1980), 241–257
MARCELLINI P., SBORDONE C., On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl., 62 (1983), 1–9
MEYERS N., An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189–206
MEYERS N., Quasiconvexity and lower semicontinuity of multiple integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125–149
MORREY C.B., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25–53
MORREY C.B., Multiple integrals in the calculus of variations, 1966, Springer-Verlag, Berlin
MURAT F., Compacité par compensation II, Proc. Inter. Meet. on Recent Meth. in Nonlinear Anal., De Giorgi, Magenes, Mosco Edit., Pitagora Bologna, 1978, 245–256
RESHETNYAK Y.G., General theorems on semicontinuity and on convergence with a functional, Sibirskii Math. J., 8 (1967), 1051–1069
RESHETNYAK Y.G., Stability theorems for mappings with bounded excursion, Sibirskii Math. J., 9 (1968), 667–684
SBORDONE C., Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa, 2 (1975), 617–638
SERRIN J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139–167
BALL J. M., MURAT F., W1,p-quasiconvexity and variational problems for multiple integrals, to appear
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marcellini, P. Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math 51, 1–28 (1985). https://doi.org/10.1007/BF01168345
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168345