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Lower semicontinuity of multiple integrals and the Biting Lemma

  • J. M. Ball (a1) and K.-W. Zhang (a2)


Weak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.



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1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125145.
2Balder, E. J.. A general approach to lower semicontinuity and lower closure in optimal control theory. SIAMJ. Control Optim. 22 (1984), 570597.
3Balder, E. J.. On infinite-horizon lower closure results for optimal control. Ann. Mat. Pura Appl. (4) (to appear).
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal 63 (1977), 337403.
5Ball, J. M.. Constitutive inequalities and existence theorems in nonlinearelasticity. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 1, ed. Knops, R. J. (London: Pitman, 1977).
6Ball, J. M.. Remarks on the paper ‘Basic Calculus of Variations’. Pacific J. Math. 116 (1985), 710.
7Ball, J. M.. A version of the fundamental theorem of Young measures. In Partial Differential Equations and Continuum Models of Phase Transitions, eds Rascle, M., Serre, D., Slemrod, M., Lecture Notes in Physics, Vol. 344, pp 207215 (Berlin: Springer, 1989).
8Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981), 135174.
9Ball, J. M. and Murat, F.. Wl.p-Quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225253.
10Ball, J. M. and Murat, F.. Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 (1989).
11Ball, J. M. and Murat, F.. In preparation.
12Berliocchi, H. and Lasry, J. M.. Integrandes normales et mesures parametres en calcul desvariations. Bull. Soc. Math. France 101 (1973), 129184.
13Brooks, J. K. and Chacon, R. V.. Continuity and compactness of measures. Adv. in Math. 37 (1980), 1626.
14Cesari, L.. Optimization–Theory and Applications, Problems with Ordinary Differential Equations (Berlin: Springer, 1983).
15Coifman, R. R., Lions, P.-L., Meyer, Y., Semmes, S.. Compacité par compensation et espaces de Hardy, Comptes Rendus Acad. Sci. Paris, to appear.
16Eisen, G.. A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals. Manuscripta Math. 27 (1979), 7379.
17Ioffe, A. D.. On lower semicontinuity of integral functions, I and II. SIAM J. Control Optim. 15 (1977), 521538; 991–1000.
18Lin, P.. Maximization of the entropy for an elastic body free of surface traction, Arch. Rat. Mech. Anal., to appear.
19Morrey, C. B.. Multiple Integrals in the Calculus of Variations (New York: Springer, 1966).
20Müller, S.. A surprising higher integrability property of mappingswith positive determinant. Bull. Amer. Math. Soc. (to appear).
21Murat, F.. A survey on compensated compactness. In Contributions to Modern Calculus of Variations, ed. Cesari, L. (Harlow: Longman, 1987).
22Reshetnyak, Y. G.. Stability theorems for mappings with bounded excursion. Siberian Math. J. 9 (1968), 499512.
23Serre, D.. Formes quadratiques et calcul des variations. J. Math. Pures Appl. 62 (1983), 177196.
24Slaby, M.. Strong convergence of vector-valued pramarts and subpramarts. Probab. Math. Statist. 5 (1985), 187196.
25Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, ed. Knops, R. J. (London: Pitman, 1979).
26Tartar, L.. The compensated method applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, NATO ASI Series, Vol. C 111, ed. Ball, J. M., pp. 263285 (Amsterdam: Reidel, 1982).
27Tartar, L.. Estimations fines des coefficients homogénisés. In Ennio DeGiorgi's Colloquium, ed. Kree, P., pp. 168187. Pitman Research Notes in Mathematics 125 (London: Pitman, 1985).
28Terpstra, F. J.. Die darstellung biquadratischer formen summen von quadraten mit anwendung auf variationsrechnung. Math. Ann. 116 (1988), 166180.
29Zhang, K.. Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Lineaire (to appear)

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Lower semicontinuity of multiple integrals and the Biting Lemma

  • J. M. Ball (a1) and K.-W. Zhang (a2)


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