Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. (English) Zbl 0573.49010

The author studies the semicontinuity of the following functional in the calculus of variations: \(F(u)=\int_{G}f(x,u,Du)dx\) where u is a vector- valued function. The function f(x,s,\(\xi)\) is assumed to verify the Carathéodory property and to be quasi-convex in Morrey’s sense. Growth conditions for f are given to ensure that F is sequentially lower semicontinuous in the weak topology of \(H^{1,p}(G,R^ N)\). The proof is based on some interesting approximation results for f. In particular, it is possible to approximate f with a non decreasing sequence of quasi- convex functions \(f_ K\), which are convex and independent of (x,s) for \(| \xi | \geq K\). Finally, polyconvex functionals in Ball’s sense are considered and some counterexamples are given.
Reviewer: E.Mascolo


49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
54C08 Weak and generalized continuity
41A30 Approximation by other special function classes
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI EuDML


[1] ACERBI E., BUTTAZZO G., FUSCO N., Semicontinuity and relaxation for integral depending on vector-valued functions, J. Math. Pures Appl., 62 (1983), 371-387 · Zbl 0481.49013
[2] ACERBI E., FUSCO N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., to appear · Zbl 0565.49010
[3] ANTMAN S.S., The influence of elasticity on analysis: modern developments, Bull. Amer. Math. Soc., 9 (1983), 267-291 · Zbl 0533.73001
[4] ATTOUCH H., SBORDONE C., Asymptotic limits for perturbed functionals of calculus of variations, Ricerche Mat., 29 (1980), 85-124 · Zbl 0453.49007
[5] BALL J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337-403 · Zbl 0368.73040
[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), 135-175 · Zbl 0459.35020
[7] DACOROGNA B., Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal., 46 (1982), 102-118 · Zbl 0547.49003
[8] DACOROGNA B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Math., 922 (1982), Springer-Verlag, Berlin · Zbl 0484.46041
[9] DAL MASO G., MODICA L., A general theory of variational functionals, Topics in Func. Anal. 1980-81, Quaderno Scuola Norm. Sup. Pisa, 1981, 149-221
[10] DE GIORGI E., Teoremi di semicontinuit? nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968-69
[11] DE GIORGI E., Sulla convergenza di alcune successioni di integrali del tipo dell’area, Rendiconti Mat., 8 (1975), 277-294 · Zbl 0316.35036
[12] EISEN G., A counterexample for some lower semicontinuity results, Math. Z., 162 (1978), 241-243 · Zbl 0379.49011
[13] EISEN G., A selection lemma for sequence of measurable sets, and lower semicontinuity of multiple integrals, Manuscripta Math., 27 (1979), 73-79 · Zbl 0404.28004
[14] EKELAND I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474 · Zbl 0441.49011
[15] EKELAND I., TEMAM R., Convex analysis and variational problems, North Holland, 1976 · Zbl 0322.90046
[16] FUSCO N., Quasi convessit? e semicontinuit? per integrali multipli di ordine superiore, Ricerche Mat., 29 (1980), 307-323 · Zbl 0508.49012
[17] GIAQUINTA M., GIUSTI E., On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31-46 · Zbl 0494.49031
[18] GIAQUINTA M., GIUSTI E., Quasi-minima, 1983, preprint · Zbl 0513.49003
[19] MARCELLINI P., Some problems of semicontinuity and of ? -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.
[20] MARCELLINI P., Quasiconvex quadratic forms in two dimensions, Appl. Math. Optimization, 11 (1984), 183-189 · Zbl 0567.49007
[21] MARCELLINI P., SBORDONE C., Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4 (1980), 241-257 · Zbl 0537.49002
[22] MARCELLINI P., SBORDONE C., On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl., 62 (1983), 1-9 · Zbl 0516.49011
[23] MEYERS N., An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206 · Zbl 0127.31904
[24] MEYERS N., Quasiconvexity and lower semicontinuity of multiple integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125-149 · Zbl 0166.38501
[25] MORREY C.B., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53 · Zbl 0046.10803
[26] MORREY C.B., Multiple integrals in the calculus of variations, 1966, Springer-Verlag, Berlin · Zbl 0142.38701
[27] MURAT F., Compacit? par compensation II, Proc. Inter. Meet. on Recent Meth. in Nonlinear Anal., De Giorgi, Magenes, Mosco Edit., Pitagora Bologna, 1978, 245-256
[28] RESHETNYAK Y.G., General theorems on semicontinuity and on convergence with a functional, Sibirskii Math. J., 8 (1967), 1051-1069
[29] RESHETNYAK Y.G., Stability theorems for mappings with bounded excursion, Sibirskii Math. J., 9 (1968), 667-684
[30] SBORDONE C., Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa, 2 (1975), 617-638
[31] SERRIN J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167 · Zbl 0102.04601
[32] BALL J. M., MURAT F., W1,p-quasiconvexity and variational problems for multiple integrals, to appear · Zbl 0549.46019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.