On lower semicontinuity of multiple integrals
Kałamajska, Agnieszka
Colloquium Mathematicae, Tome 72 (1997), p. 71-78 / Harvested from The Polish Digital Mathematics Library

We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional ΩF(x,u,u)dx. The proofs are based on arguments from the theory of Young measures.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:210502
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     author = {Agnieszka Ka\l amajska},
     title = {On lower semicontinuity of multiple integrals},
     journal = {Colloquium Mathematicae},
     volume = {72},
     year = {1997},
     pages = {71-78},
     zbl = {0893.49009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv74i1p71bwm}
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Kałamajska, Agnieszka. On lower semicontinuity of multiple integrals. Colloquium Mathematicae, Tome 72 (1997) pp. 71-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv74i1p71bwm/

[000] [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125-145. | Zbl 0565.49010

[001] [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. | Zbl 0642.49007

[002] [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.

[003] [4] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. | Zbl 0368.73040

[004] [5] J. M. Ball and F. Murat, Remarks on Chacon's Biting Lemma, Proc. Amer. Math. Soc. 107 (1989), 655-663. | Zbl 0678.46023

[005] [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. | Zbl 0716.49011

[006] [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. | Zbl 0835.46025

[007] [8] B. Bojarski and P. Hajłasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77-92. | Zbl 0810.46030

[008] [9] A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, ibid. 20 (1961), 171-225. | Zbl 0099.30103

[009] [10] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin, 1989. | Zbl 0703.49001

[010] [11] M. Esteban, A direct variational approach to Skyrme's model for meson fields, Comm. Math. Phys. 105 (1986), 571-591. | Zbl 0621.58035

[011] [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.

[012] [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976. | Zbl 0322.90046

[013] [14] A. D. Ioffe, On lower semicontinuity of integral functionals, I, II, SIAM J. Control Optim. 15 (1977), 521-538, 991-1000. | Zbl 0361.46037

[014] [15] D. Kinderlehrer and P. Pedregal, Characterisation of Young measures generated by gradients, Arch. Rational Mech. Anal. 115 (1991), 329-365. | Zbl 0754.49020

[015] [16] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. (to appear). | Zbl 0808.46046

[016] [17] J. Kristensen, Lower semicontinuity of variational integrals, Ph.D. Thesis, Mathematical Institute, The Technical University of Denmark, 1994.

[017] [18] F. C. Liu, A Luzin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645-651. | Zbl 0368.46036

[018] [19] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), 1-28. | Zbl 0573.49010

[019] [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. | Zbl 0166.38501

[020] [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. | Zbl 0592.41031

[021] [22] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.

[022] [23] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin, 1990.

[023] [24] K. Zhang, Biting theorems for Jacobians and their applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 345-365. | Zbl 0717.49012