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1997 | 74 | 1 | 71-78
Tytuł artykułu

On lower semicontinuity of multiple integrals

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
74
Numer
1
Strony
71-78
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-04-01
poprawiono
1996-11-27
Twórcy
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv74i1p71bwm
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