We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional $\int_{Ω} F(x,u,∇u)dx$. The proofs are based on arguments from the theory of Young measures.
Institute of Mathematics, University of Warsaw, Banacha 2, 00-097 Warszawa, Poland
Bibliografia
[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125-145.
[2] L. Ambrosio, New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Mat. Natur. 105 (1987), 1-42.
[3] 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, Berlin, 1989.
[4] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403.
[5] J. M. Ball and F. Murat, Remarks on Chacon's Biting Lemma, Proc. Amer. Math. Soc. 107 (1989), 655-663.
[6] J. M. Ball and K. W. Zhang, Lower semicontinuity of multiple integrals and the Biting Lemma, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 367-379.
[7] B. Bojarski, Remarks on some geometric properties of Sobolev mappings, in: Functional Analysis and Related Topics, S. Koshi (ed.), World Scientific, Singapore, 1991, 65-76.
[8] B. Bojarski and P. Hajłasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77-92.
[9] A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, ibid. 20 (1961), 171-225.
[10] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin, 1989.
[11] M. Esteban, A direct variational approach to Skyrme's model for meson fields, Comm. Math. Phys. 105 (1986), 571-591.
[12] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CMBS Regional Conf. Ser. in Math. 74, Amer. Math. Soc., Providence, R.I., 1990.
[13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
[14] A. D. Ioffe, On lower semicontinuity of integral functionals, I, II, SIAM J. Control Optim. 15 (1977), 521-538, 991-1000.
[15] D. Kinderlehrer and P. Pedregal, Characterisation of Young measures generated by gradients, Arch. Rational Mech. Anal. 115 (1991), 329-365.
[16] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. (to appear).
[17] J. Kristensen, Lower semicontinuity of variational integrals, Ph.D. Thesis, Mathematical Institute, The Technical University of Denmark, 1994.
[18] F. C. Liu, A Luzin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645-651.
[19] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), 1-28.
[20] N. G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational functionals of any order, Trans. Amer. Math. Soc. 119 (1965), 125-149.
[21] J. Michael and W. Ziemer, A Lusin type approximation of Sobolev functions by smooth functions, in: Contemp. Math. 42, Amer. Math. Soc., 1985, 135-167.
[22] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.
[23] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin, 1990.
[24] K. Zhang, Biting theorems for Jacobians and their applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 345-365.
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