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Lower semicontinuity of multiple integrals and the Biting Lemma

Published online by Cambridge University Press:  14 November 2011

J. M. Ball  and
K.-W. Zhang
J. M. Ball
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.
K.-W. Zhang
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.; Department of Mathematics, Peking University, Beijing 100 871, China
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Synopsis

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.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 367 - 379
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125–145.CrossRefGoogle Scholar
2Balder, E. J.. A general approach to lower semicontinuity and lower closure in optimal control theory. SIAMJ. Control Optim. 22 (1984), 570–597.CrossRefGoogle Scholar
3Balder, E. J.. On infinite-horizon lower closure results for optimal control. Ann. Mat. Pura Appl. (4) (to appear).Google Scholar
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal 63 (1977), 337–403.CrossRefGoogle Scholar
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).Google Scholar
6Ball, J. M.. Remarks on the paper ‘Basic Calculus of Variations’. Pacific J. Math. 116 (1985), 7–10.CrossRefGoogle Scholar
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 207–215 (Berlin: Springer, 1989).Google Scholar
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), 135–174.CrossRefGoogle Scholar
9Ball, J. M. and Murat, F.. Wl.p-Quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
10Ball, J. M. and Murat, F.. Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 (1989).Google Scholar
11Ball, J. M. and Murat, F.. In preparation.Google Scholar
12Berliocchi, H. and Lasry, J. M.. Integrandes normales et mesures parametres en calcul desvariations. Bull. Soc. Math. France 101 (1973), 129–184.CrossRefGoogle Scholar
13Brooks, J. K. and Chacon, R. V.. Continuity and compactness of measures. Adv. in Math. 37 (1980), 16–26.CrossRefGoogle Scholar
14Cesari, L.. Optimization–Theory and Applications, Problems with Ordinary Differential Equations (Berlin: Springer, 1983).Google Scholar
15Coifman, R. R., Lions, P.-L., Meyer, Y., Semmes, S.. Compacité par compensation et espaces de Hardy, Comptes Rendus Acad. Sci. Paris, to appear.Google Scholar
16Eisen, G.. A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals. Manuscripta Math. 27 (1979), 73–79.CrossRefGoogle Scholar
17Ioffe, A. D.. On lower semicontinuity of integral functions, I and II. SIAM J. Control Optim. 15 (1977), 521–538; 991–1000.CrossRefGoogle Scholar
18Lin, P.. Maximization of the entropy for an elastic body free of surface traction, Arch. Rat. Mech. Anal., to appear.Google Scholar
19Morrey, C. B.. Multiple Integrals in the Calculus of Variations (New York: Springer, 1966).CrossRefGoogle Scholar
20Müller, S.. A surprising higher integrability property of mappingswith positive determinant. Bull. Amer. Math. Soc. (to appear).Google Scholar
21Murat, F.. A survey on compensated compactness. In Contributions to Modern Calculus of Variations, ed. Cesari, L. (Harlow: Longman, 1987).Google Scholar
22Reshetnyak, Y. G.. Stability theorems for mappings with bounded excursion. Siberian Math. J. 9 (1968), 499–512.CrossRefGoogle Scholar
23Serre, D.. Formes quadratiques et calcul des variations. J. Math. Pures Appl. 62 (1983), 177–196.Google Scholar
24Slaby, M.. Strong convergence of vector-valued pramarts and subpramarts. Probab. Math. Statist. 5 (1985), 187–196.Google Scholar
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).Google Scholar
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. 263–285 (Amsterdam: Reidel, 1982).Google Scholar
27Tartar, L.. Estimations fines des coefficients homogénisés. In Ennio DeGiorgi's Colloquium, ed. Kree, P., pp. 168–187. Pitman Research Notes in Mathematics 125 (London: Pitman, 1985).Google Scholar
28Terpstra, F. J.. Die darstellung biquadratischer formen summen von quadraten mit anwendung auf variationsrechnung. Math. Ann. 116 (1988), 166–180.CrossRefGoogle Scholar
29Zhang, K.. Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Lineaire (to appear)Google Scholar