Free access
Issue
ESAIM: COCV
Volume 7, 2002
Page(s) 443 - 470
DOI http://dx.doi.org/10.1051/cocv:2002063
Published online 15 September 2002
  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational. Mech. Anal. 86 (1984) 125-145. [CrossRef] [MathSciNet]
  2. E. Acerbi and N. Fusco, An approximation lemma for W1,pfunctions, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems, edited by J.M. Ball. Heriot-Watt University, Oxford (1988).
  3. E. Anzelotti, S. Baldo and D. Percivale, Dimensional reduction in variational problems, asymptotic developments in Formula -convergence, and thin structures in elasticity. Asymptot. Anal. 9 (1994) 61-100.
  4. E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570-598. [CrossRef] [MathSciNet]
  5. J.M. Ball, A version of the fundamental theorem for Young mesures, in PDE's and Continuum Models of Phase Transitions, edited by M. Rascle, D. Serre and M. Slemrod. Springer-Verlag, Berlin, Lecture Notes in Phys. 344 (1989) 207-215. [CrossRef]
  6. H. Berliocchi and J.-M. Lasry, Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. [MathSciNet]
  7. K. Bhattacharya and A. Braides, Thin films with many small cracks. Preprint (2000).
  8. K. Bhattacharya, I. Fonseca and G. Francfort, An asymptotic study of the debonding of thin films. Arch. Rational. Mech. Anal. 161 (2002) 205-229. [CrossRef]
  9. K. Bhattacharya and R.D. James, A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999) 531-576. [CrossRef] [MathSciNet]
  10. A. Braides, Private communication.
  11. A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367-1404. [MathSciNet]
  12. A. Braides and I. Fonseca, Brittle thin films. Appl. Math. Optim. 44 (2001) 299-323. [CrossRef] [MathSciNet]
  13. S. Conti, I. Fonseca and G. Leoni, A Formula -convergence result for the two-gradient theory of phase transitions, Preprint 01-CNA-008. Center for Nonlinear Analysis, Carnegie Mellon University (2001). Comm. Pure Applied Math. (to appear).
  14. B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989).
  15. I. Fonseca and G. Francfort, On the inadequacy of scaling of linear elasticity for 3D-2D asymptotics in a nonlinear setting. J. Math. Pures Appl. 80 (2001) 547-562. [CrossRef] [MathSciNet]
  16. I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations with Applications to Nonlinear Continuum Physics. Springer-Verlag (to appear).
  17. I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. [CrossRef] [MathSciNet]
  18. D.D. Fox, A. Raoult and J.C. Simo, A justification of nonlinear properly invariant plate theories. Arch. Rational. Mech. Anal. 124 (1993) 157-199. [CrossRef] [MathSciNet]
  19. T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses. Arch. Rational. Mech. Anal. 119 (1992) 129-143. [CrossRef] [MathSciNet]
  20. D. Kinderlehrer and P. Pedregal, Characterizations of Young mesures generated by gradients. Arch. Rational. Mech. Anal. 115 (1991) 329-365. [CrossRef] [MathSciNet]
  21. D. Kinderlehrer and P. Pedregal, Gradient Young mesures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. [CrossRef] [MathSciNet]
  22. J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mathematical Institute, Technical University of Denmark, Mat-Report No. 1994-34 (1994).
  23. J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653-710. [CrossRef] [MathSciNet]
  24. H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. [MathSciNet]
  25. H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Rational. Mech. Anal. 154 (2000) 101-134. [CrossRef] [MathSciNet]
  26. F.C. Liu, A Luzin type property of Sobolev functions. Indiana Univ. Math. J. 26 (1997) 645-651.
  27. P. Pedregal, Parametrized mesures and Variational Principles. Birkhäuser, Boston (1997).
  28. E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970).
  29. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, edited by R. Knops. Longman, Harlow, Pitman Res. Notes Math. Ser. 39 (1979) 136-212.
  30. L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, edited by J.M. Ball. Riedel (1983).
  31. L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lecture Notes in Phys. 195 (1994) 384-412. [CrossRef]
  32. Y.C. Shu, Heterogeneous thin films of martensitic materials. Arch. Rational. Mech. Anal. 153 (2000) 39-90. [CrossRef]
  33. L.C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Lettres de Varsovie, Classe III 30 (1937) 212-234.
  34. L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders (1969).
  35. W.P. Ziemer, Weakly Differentiable Functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, Berlin (1989).