A-quasiconvexity, lower semicontinuity, and young measures

SIAM Journal on Mathematical Analysis

Volumn 30, Issue 6, 1999, Pages 1355-1390

  Fonseca, Irene ^a   Müller, Stefan ^b  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

A quasiconvexity; Equi integrability young measure; Lower semicontinuity

Indexed keywords

EID: 0033249123     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141098339885     Document Type: Article

References (47)

1. E. ACERBI AND N. FUSCO, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), pp. 125-145.
2. J. J. ALIBERT AND G. BOUCHITTÉ, Non uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), pp. 129-147.
3. m) of quasi-convex integrals, J. Funct. Anal., 109 (1992), pp. 76-97.
4. E. J. BALDER, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), pp. 570-598.
5. J. M. BALL, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977), pp. 337-403.
6. J. M. BALL, A version of the fundamental theorem for Young measures, in PDE's and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., Lecture Notes in Phys. 344, Springer-Verlag, Berlin, 1989, pp. 207-215.
7. J. M. BALL AND R. D. JAMES, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), pp. 13-52.
8. J. M. BALL AND R. D. JAMES, Proposed experimental tests of a theory of fine micro structure and the two well problem, Philos. Trans. Roy. Soc. London, 338 (1992), pp. 389-450.
9. l,p-quasiconvexity and vanational problems for multiple integrals, J. Funct. Anal, 58 (1984), pp. 222-253.
10. J. M. BALL AND F. MURAT, Remarks on rank-one convexity and quasiconvexity, in Ordinary and Partial Differential Equations, Vol. III, B. D. Sleeman and R. J. Jarvis, eds., Longman, Harlow, UK, 1991, pp. 25-37.
11. H. BERLIOCCHI AND J.-M. LASRY, Intégrands normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France, 101 (1973), pp. 129-184.
12. G. BOUCHITTÉ, I. FONSECA, AND L. MASCARENHAS, A global method for relaxation, Arch. Rational Mech. Anal, 145 (1998), pp. 55-98.
13. M. CHIPOT AND D. KINDERLEHRER, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., (1998), pp. 237-277.
14. B. DACOROGNA, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals, Lecture Notes in Math. 922, Springer-Verlag, New York, 1982.
15. B. DACOROGNA, Quasiconvexity and relaxation of non convex variational problems, J. Funct. Anal., 46 (1982), pp. 102-118.
16. A. DE SIMONE, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal., 125 (1993), pp. 99-143.
17. R. J. DIPERNA, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), pp. 27-70.
18. R. J. DIPERNA, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), pp. 383-42.
19. R. J. DIPERNA AND A. J. MAJDA, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), pp. 667-689.
20. L. C. EVANS, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conf. Ser. in Math. 74, AMS, Providence, RI, 1990.
21. I. FONSECA, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl., 67 (1988), pp. 175-195.
22. 1, SIAM J. Math. Anal., 23 (1992), pp. 1081-1098.
23. p) for integrands f (x, u, ∇u), Arch. Rational Mech. Anal., 123 (1993), pp. 1-49.
24. I. FONSECA, S. MÜLLER, AND P. PEDREGAL, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal., 29 (1998), pp. 736-756.
25. P. GERARD, Compacité par compensation et regularité 2-microlocale, Séminaire Eq. aux Dér. Part., Ecole Polytechnique, Palaiseau, exp VI, 1988-89.
26. P. GERARD, Microlocal defect measure, Comm. Partial Differential Equations, 16 (1989), pp. 1761-1794.
27. 2, Phil. Mag. B, 68 (1993), pp. 237-274.
28. D. KINDERLEHRER AND P. PEDREGAL, Characterizations of Young measures generated by gradients, Arch. Rational Mech. Anal., 115 (1391), pp. 329-365.
29. D. KINDERLEHRER AND P. PEDREGAL, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4 (1994), pp. 59-90.
30. J. KRISTENSEN, Finite Functional and Young Measures Generated by Gradients of Sobolev Functions, Mat-Report 1994-34, Mathematical Institute, Technical University of Denmark, Lyngby, Denmark, 1994.
31. P. MARCELLINI, Approximation of quasiconvex functions and semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), pp. 1-28.
32. C. B. MORREY, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.
33. S. MÜLLER, Rank-One Convexity Implies Quasiconvexity on Diagnol Matrices, Preprint 23/1999, Max-Planck Institute for Mathematics in the Sciences, Leiptig, Germany, 1999.
34. F. MURAT, Compacité par compensation : condition necessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), pp. 68-102.
35. P. PEDREGAL, Parametrized Measures and Variational Principles, Birkhäuser, Boston, 1997.
36. M. E. SCHONBECK, Convergence of solutions to non-linear dispersive equations, Comm. in Partial Differential Equations, 7 (1982), pp. 959-1000.
37. J. R. SCHULENBERGER AND C. H. WILCOX, Coerciveness inequalities for nonelliptic systems of partial differential equations, Annali di Matematica, 88 (1971), pp. 229-305.
38. E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
39. E. M. STEIN AND G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.
40. M. A. SYCHEV, A new approach to Young measure theory, relaxation and convergence in energy, Ann. IHP Anal. Non Linéaire, Inst. H. Poincaré, to appear.
41. L. TARTAR, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, R. Knops, ed., Vol. IV, Pitman Res. Notes Math. 39, Longman, Harlow, UK, 1979, pp. 136-212.
42. L. TARTAR, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, J. M. Ball, ed., D. Riedel, Dordrecht, 1983.
43. L. TARTAR, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires, in Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Phys. 195, Springer-Verlag, Berlin, New York, 1984, pp. 384-412.
44. L. TARTAR, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), pp. 193-230.
45. L. TARTAR, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures, in Developments in Partial Differential Equations and Applications to Mathematical Physics, G. Buttazzo, G. P. Galdi, and L. Zanghirati, eds., Plenum, New York, 1991.
46. L. TARTAR, Some remarks on separately convex functions, in Microstructure and Phase Transitions, D. Kinderlehrer, R. D. James, M. Luskin, and J. L. Ericksen, eds., IMA Vol. Math. Appl. 54, Springer-Verlag, New York, 1993, pp. 191-204.
47. L. C. YOUNG, Lectures on Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969.