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Non convex integrals of the Calculus of Variations

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Methods of Nonconvex Analysis

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References

  1. E. ACERBI-G. BUTTAZZO, Semicontinuous envelopes of polyconvex integrals, Proc. R. Soc. Edinb., 96A, (1984), 51–54.

    Article  MathSciNet  Google Scholar 

  2. E. ACERBI-G. BUTTAZZO-N. FUSCO, Semicontinuity and relaxation for integrals depending on vector-valued functions, J. Math. Pures Appl., 62 (1983), 371–387.

    MathSciNet  MATH  Google Scholar 

  3. E. ACERBI-N. FUSCO, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125–145.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. ACERBI-N. FUSCO, A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal., 99 (1987), 261–281.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. AMBROSIO, New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL, 11 (1987), 1–42.

    MathSciNet  MATH  Google Scholar 

  6. S. ANTMAN, The influence of elasticity on analysis: modern developments, Bull. Amer. Math. Soc., 9 (1983), 267–291.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. ANZELLOTTI-G. BUTTAZZO-G. DAL MASO, Dirichlet problems for demicoercive functionals, Nonlinear Anal., 10 (1986), 603–613.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. ATTOUCH, Variational Convergence of Functional and operators, Pitman, London, 1984.

    MATH  Google Scholar 

  9. G. AUBERT, A counterexample of a rank one convex function which is not polyconvex in the case n=2, Proc. R.Soc. Edinb., 106A (1987), 237–240.

    Article  MathSciNet  Google Scholar 

  10. G. AUBERT-R. TAHRAOUI, Théorèmes d'existence pour des problèmes du calcul des variations, J. Differential Eq., 33 (1979), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. AUBERT-R. TAHRAOUI, Sur la minimisation d'une fonctionelle non convexe, non différentiable en dimension 1, Boll. Un. Mat. Ital., 17 (1980), 244–258.

    MathSciNet  MATH  Google Scholar 

  12. G. AUBERT-R. TAHRAOUI, Théorèmes d'existence en optimimisation non convexe, Appl. Anal., 18 (1984), 75–100.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. AUBERT-R. TAHRAOUI, Sur la faible fermature de certain ensembles de constraintes en élasticité non-linéaire plane, Arch. Rational Mech. Anal., 97 (1987), 33–58.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.M.BALL, On the calculus of variations and sequentially weakly continuous maps, Ordinary and Partial Differential Equat., Lecture Notes in Math. n. 564, Springer, 1976, 13–25.

    Google Scholar 

  15. J.M. BALL, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977), 337–403.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.M. BALL, Discontinuous equilibrium solutions and cavitation in non-linear elasticity, Phil. Trans. R. Soc. London, 306 (1982), 557–611.

    Article  MATH  Google Scholar 

  17. J.M. BALL-J.C. CURRIE-P.J. OLVER, Null lagrangians, weak continuity and variational problems of arbitrary order, J. Func. Analysis, 41 (1981), 135–175.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.M.BALL-R.J.KNOPS-J.E.MARSDEN, Two examples in nonlinear elasticity, Journée d'analyse non linéaire, Lecture Notes in Math. 665, Springer, 41–48.

    Google Scholar 

  19. J.M. BALL-J.E. MARSDEN, Quasiconvexity at the boundary, positivity of the second variation and elastic stability, Arch. Rational Mech. Anal., 86 (1984), 251–277.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.M. BALL-V.J. MIZEL, One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal., 90 (1985), 325–388.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.M. BALL-F. MURAT, W1,p quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225–253.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. BENSOUSSAN-J.L. LIONS-G. PAPANICOLAOU, Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  23. L.D. BERKOWITZ, Lower semicontinuity of integral functionals, Trans. Am. Math. Soc. 192 (1974), 51–57.

    Article  MathSciNet  Google Scholar 

  24. F.BETHUEL-H.BREZIS-J.M.CORON, Relaxed energies for harmonic maps, preprint Université Pierre et Marie Curie, 1989.

    Google Scholar 

  25. P.J. BLATZ-W.L. KO, Application of finite elastic theory to the deformation of rubbery materials, Trans. Soc. Rheology, 6 (1962), 223–251.

    Article  Google Scholar 

  26. L. BOCCARDO-G. BUTTAZZO, Quasilinear elliptic equations with discontinuous coefficients, Atti Accad. Naz. Lincei, 82 (1988), 21–28.

    MathSciNet  MATH  Google Scholar 

  27. L. BOCCARDO-B. DACOROGNA, Monotonicity of certain differential operators in divergence form, Manuscripta Math., 64 (1989), 253–260.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. BONI, Un teorema di semicontinuità inferiore, Rend. Circ. Mat. Palermo, 25 (1976), 53–66.

    Article  MathSciNet  MATH  Google Scholar 

  29. B.BOTTERON-P.MARCELLINI, A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations, preprint Università di Firenze, 1989.

    Google Scholar 

  30. G. BOTTARO-P. OPPEZZI, Rappresentazione con integrali multipli di funzionali dipendenti da funzioni a valori in uno spazio di Banach, Ann. Mat. Pura Appl., 139 (1985), 191–225.

    Article  MathSciNet  Google Scholar 

  31. H. BREZIS, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), 9–23.

    Article  MathSciNet  Google Scholar 

  32. G.BUTTAZZO, Semicontinuity, relaxation, and integral representation in the calculus of variations, Research Notes in Math., Longman, to appear.

    Google Scholar 

  33. G. BUTTAZZO-G. DAL MASO, A characterization of nonlinear functionals on Sobolev spaces which admit an integral representation with a Carathéodory integrand, J. Math. Pures Appl., 64 (1985), 337–361.

    MathSciNet  MATH  Google Scholar 

  34. G. BUTTAZZO-G. DAL MASO, Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), 515–532.

    Article  MathSciNet  MATH  Google Scholar 

  35. R. CACCIOPPOLI-G. SCORZA DRAGONI, Necessità della condizione di Weierstrass per la semicontinuità di un integrale doppio sopra una data superficie, Memorie Acc. d'Italia, 9 (1938), 251–268.

    MATH  Google Scholar 

  36. L.CARBONE-R.DE ARCANGELIS, Further results on Γ-convergence and lower semicontinuity of integral functionals depending on vector-valued functions, preprint Università di Salerno, 1989.

    Google Scholar 

  37. L. CARBONE-C. SBORDONE, Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl., 122 (1979), 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  38. M. CARRIERO-A. LEACI-E. PASCALI, On the semicontinuity and the relaxation for integrals with respect to the Lebesgue measure added to integrals with respect to a Radon measure, Ann. Mat. Pura Appl., 149 (1987), 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  39. E.CASADIO TARABUSI, An algebraic characterization of quasi-convex functions, preprint.

    Google Scholar 

  40. A.CELLINA-G.COLOMBO, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. Henri Poincaré, Analyse non Linéaire, to appear.

    Google Scholar 

  41. L. CESARI, Lower semicontinuity and lower closure theorems without seminormality conditions, Ann. Mat. Pura Appl., 98 (1974), 381–397.

    Article  MathSciNet  MATH  Google Scholar 

  42. L. CESARI, An existence theorem without convexity conditions, SIAM J. Control, 12 (1974), 319–331.

    Article  MathSciNet  MATH  Google Scholar 

  43. L.CESARI, Optimization — Theory and applications, Appl. of Math. 17, Springer-Verlag, 1983.

    Google Scholar 

  44. L. CESARI-P. BRANDI-A. SALVADORI, Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions, Arch. Rational Mech. Anal., 98 (1987), 307–328.

    Article  MathSciNet  MATH  Google Scholar 

  45. P. CHARRIER-B. DACOROGNA-B. HANOUZET-P. LABORDE, An existence theorem for slightly compressible material in nonlinear elasticity, S.I.A.M. Math. Anal., 19 (1988), 70–86.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. CHIPOT-D. KINDERLEHER, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103 (1988), 237–277.

    Article  MathSciNet  MATH  Google Scholar 

  47. P.G. CIARLET, Mathematical Elasticity, vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1987.

    Google Scholar 

  48. P.G. CIARLET-J. NEČAS, Unilateral problems in nonlinear, thee-dimensional elasticity, Arch. Rational Mech. Anal., 87 (1985), 319–338.

    Article  MathSciNet  MATH  Google Scholar 

  49. F.H. CLARKE, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

    MATH  Google Scholar 

  50. J.N.CORVELLEC, Quelques remarques sur le problème de Lagrange du calcul des variations, Colloquium Math., 57 (1989), to appear.

    Google Scholar 

  51. B. DACOROGNA, A relaxation theorem and its applications to the equilibrium of gases, Arch. Rational Mech. Anal., 77 (1981), 359–386.

    Article  MathSciNet  MATH  Google Scholar 

  52. B. DACOROGNA, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lectures Notes in Math. 922, Springer-Verlag, Berlin, 1982.

    Book  MATH  Google Scholar 

  53. B. DACOROGNA, Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Func. Analysis, 46 (1982), 102–118.

    Article  MathSciNet  MATH  Google Scholar 

  54. B.DACOROGNA, Direct methods in the calculus of variations, Applied Math. Sciences 78, Springer-Verlag, 1989.

    Google Scholar 

  55. B. DACOROGNA-N. FUSCO, Semicontinuité des fonctionnelles avec contraites du type “det Δu > 0”, Boll. Un. Mat. Ital., 4-B (1985), 179–189.

    MathSciNet  MATH  Google Scholar 

  56. B. DACOROGNA-P. MARCELLINI, A counterexample in the vectorial calculus of variations, Material Instabilities in Continuum Mechanics, edited by J.M. Ball, Calderon Press, Oxford, 1988, 77–83.

    Google Scholar 

  57. B.DACOROGNA-J.MOSER, On a partial differential equation involving the Jacobian determinant, Ann. Inst. Henri Poincaré, Analyse non Linéaire, to appear.

    Google Scholar 

  58. G. DAL MASO, Integral representation on BV(Ω) of Γ-limits of variational integrals, Manuscripta Math., 30 (1980), 387–416.

    Article  MathSciNet  MATH  Google Scholar 

  59. G.DAL MASO-L.MODICA, A general theory of variational functionals, Topics in Funct. Analysis 1980–81, Quaderni della Scuola Normale Sup. di Pisa, 1982.

    Google Scholar 

  60. G. DAL MASO-L. MODICA, Integral functionals determined by their minima, Rend. Sem. Mat. Univ. Padova, 76 (1986), 255–267.

    MathSciNet  MATH  Google Scholar 

  61. E. DE GIORGI, Teoremi di semicontinuità nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968–1969.

    Google Scholar 

  62. E. DE GIORGI, Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rendiconti Mat., 8 (1975), 277–294.

    MathSciNet  MATH  Google Scholar 

  63. E. DE GIORGI, Some continuity and relaxation problems, E.De Giorgi Colloquium, Proceedings, Pitman Res. Notes, vol. 125, London (1985), 1–11.

    MathSciNet  MATH  Google Scholar 

  64. E. DE GIORGI-G. BUTTAZZO-G. DAL MASO, On the lower semicontinuity of certain integral functionals, Atti Accad. Naz. Lincei, 74 (1983), 274–282.

    MathSciNet  MATH  Google Scholar 

  65. I. EKELAND, Discontinuités de champs hamiltionens et existence de solutions optimales en calcul des variations, Publications Math. de l' I.H.E.S., 47 (1977), 5–32.

    Article  MathSciNet  MATH  Google Scholar 

  66. I. EKELAND, Nonconvex mimimization problems, Bull. Amer. Math. Soc., 1 (1979), 443–474.

    Article  MathSciNet  MATH  Google Scholar 

  67. I.EKELAND-R.TEMAN, Analyse convexe et problèmes variationnels, Dunod Gauthier-Villars, 1974.

    Google Scholar 

  68. L.C. EVANS, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal., 95 (1986), 227–252.

    Article  MathSciNet  MATH  Google Scholar 

  69. L.C. EVANS-R.F. GARIEPY, Blow-up, compacteness and partial regularity in the calculus of variations, Indiana U. Math. J., 36 (1987), 361–371.

    Article  MathSciNet  MATH  Google Scholar 

  70. F. FERRO, Functionals defined on functions of bounded variation in ℝ n and the Lebesgue area, SIAM J.Control Optimization, 16 (1978), 778–789.

    Article  MathSciNet  MATH  Google Scholar 

  71. N. FUSCO, Quasi convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat., 29 (1980), 307–323.

    MathSciNet  MATH  Google Scholar 

  72. N. FUSCO-J. HUTCHINSON, C 1,α partial regularity of functions minimizing quasiconvex integrals, Manuscripta Math.,54 (1985), 121–143.

    Article  MathSciNet  Google Scholar 

  73. N.FUSCO-J.HUTCHINSON, Partial regularity in problems motivated by non-linear elasticity, 1989, preprint.

    Google Scholar 

  74. N. FUSCO-G. MOSCARIELLO, L2-lower semicontinuity of functionals of quadratic type, Ann. Mat. Pura Appl., 129 (1981), 305–326.

    Article  MathSciNet  MATH  Google Scholar 

  75. V. GAVIOLI, A lower semicontinuity theorem for the integral of the calculus of variations, Atti Sem. Mat. Fis. Univ. Modena, 31 (1982), 268–284.

    MathSciNet  MATH  Google Scholar 

  76. M.GIAQUINTA, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Stud. 105, Princeton Univ. Press, 1983.

    Google Scholar 

  77. M.GIAQUINTA, Quasiconvexity, growth conditions and partial regularity, P.D.E. and Calculus of Variations, S.Hildebrandt and R.Leis editors, Lecture Notes in Math. 1357, Springer, to appear.

    Google Scholar 

  78. M. GIAQUINTA-E. GIUSTI, On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31–46.

    Article  MathSciNet  MATH  Google Scholar 

  79. M. GIAQUINTA-E. GIUSTI, Q-minima, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 1 (1984), 79–107.

    MathSciNet  MATH  Google Scholar 

  80. M. GIAQUINTA-G. MODICA, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 3 (1986), 185–208.

    MathSciNet  MATH  Google Scholar 

  81. M. GIAQUINTA-G. MODICA-J. SOUČEK, Functionals with linear growth in the calculus of variations I, Commentationes Math.Univ. Carolinae, 20 (1979), 143–172.

    MathSciNet  MATH  Google Scholar 

  82. M. GIAQUINTA-G. MODICA-J. SOUČEK, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 106 (1989), 97–159. Erratum in the same journal, to appear.

    Article  MathSciNet  MATH  Google Scholar 

  83. E. GIUSTI, Non-parametric minimal surfaces with discontinuous and thin obstacles, Arch. Rational Mech. Anal., 49 (1972), 41–56.

    Article  MathSciNet  MATH  Google Scholar 

  84. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984.

    Book  MATH  Google Scholar 

  85. H.GOLDSTINE, A history of the calculus of variations from the 17th through the 19th century, Studies in History of Math. and Phys. Sc. 5, Springer-Verlag, 1980.

    Google Scholar 

  86. C. HAMBURGER, A characterization for rank-one-convexity in two dimensions, Ricerche Mat., 36 (1987), 171–181.

    MathSciNet  MATH  Google Scholar 

  87. HONG MIN-CHUN, Existence and partial regularity in the calculus of variations, Ann. Mat. Pura Appl., 149 (1987), 311–328.

    Article  MathSciNet  Google Scholar 

  88. C.O. HORGAN-R. ABEYARATNE, A bifurcation problem for a compressible nonlinear elastic medium: growth of a microvoid, J. of Elasticity, 16 (1986), 189–200.

    Article  MathSciNet  MATH  Google Scholar 

  89. A.D. IOFFE, On lower semicontinuity of integral functionals I, SIAM J. Control Optimization, 15 (1977), 521–538.

    Article  MathSciNet  MATH  Google Scholar 

  90. D.KINDERLEHRER-E.MASCOLO, Local minima for nonconvex problems, preprint.

    Google Scholar 

  91. D. KINDERLHERER-G. STAMPACCHIA, Introduction to variational inequalities and their applications, Academic Press, New York, 1980.

    Google Scholar 

  92. R. KLOTZLER, On the existence of optimal processes, Banach Center Publications, Vol. 1, Warszawa, 1976, 125–130.

    Google Scholar 

  93. R.J. KNOPS-C.A. STUART, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal., 86 (1984), 233–249.

    Article  MathSciNet  MATH  Google Scholar 

  94. J.K. KNOWLES-E. STERNBERG, On the ellipticity of the equation of nonlinear elastostatics for a special material, J. of Elasticity, 5 (1975), 341–361.

    Article  MathSciNet  MATH  Google Scholar 

  95. R.V. KOHN-G. STRANG, Optimal design annd relaxation of variational problems I,II,III, Commu. Pure Appl. Math., 39 (1986), 113–137, 139–182, 353–377.

    Article  MATH  Google Scholar 

  96. H. LEBESGUE, Intégrale, longueur, aire, Ann. Mat. Pura Appl., 7 (1902), 231–359.

    Article  MATH  Google Scholar 

  97. J.L.LIONS-E.MAGENES, Non-homogeneous boundary value problems and applications I,II,III, Springer-Verlag, 1972 and 1973.

    Google Scholar 

  98. P.L.LIONS, Generalized solutions of Hamilton-Jacobi equations, Pitman, 1982.

    Google Scholar 

  99. P.MARCELLINI, Some problems of semicontinuity and of Γ-convergence for integrals of the calculus of variations, Recent Methods in Nonlinear Analisys, edited by De Giorgi, Magenes, Mosco, Pitagora Ed., 1979, 205–222.

    Google Scholar 

  100. P. MARCELLINI, Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità, Rendiconti di Matematica, 13 (1980), 271–281.

    MathSciNet  MATH  Google Scholar 

  101. P.MARCELLINI, A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations, Mathematical Theories of Optimization, Proceedings, edited by J.P.Cecconi and T.Zolezzi, Lecture Notes in Math., vol. 979, Springer, 1983, 216–231.

    Google Scholar 

  102. P. MARCELLINI, Some remarks on uniqueness in the calculus of variations, Collège de France Seminar, edited by H.Brézis and J.L.Lions, Research Notes in Math., vol. 84, Pitman, 1983, 148–153.

    MathSciNet  MATH  Google Scholar 

  103. P. MARCELLINI, Quasiconvex quadratic forms in two dimensions, Appl. Math. Optimization, 11 (1984), 183–189.

    Article  MathSciNet  MATH  Google Scholar 

  104. P. MARCELLINI, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  105. P. MARCELLINI, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 3 (1986), 391–409.

    MathSciNet  MATH  Google Scholar 

  106. P.MARCELLINI, Existence theorems in nonlinear elasticity, edited by J.B.Hiriart-Urruty, North-Holland, 1986, 241–247.

    Google Scholar 

  107. P.MARCELLINI, The stored-energy for some discontinuous deformations in nolinear elasticity, Essays in honor of E.De Giorgi, Vol.II, edited by F.Colombini et al., Birkhäuser, 1989, 767–786.

    Google Scholar 

  108. P. MARCELLINI-C. SBORDONE, Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4 (1980), 241–257.

    Article  MathSciNet  MATH  Google Scholar 

  109. P. MARCELLINI-C. SBORDONE, On the existence of minima of multiple integrals of the calculus of variations, J.Math. Pures Appl., 62 (1983), 1–9.

    MathSciNet  MATH  Google Scholar 

  110. E. MASCOLO, Some remarks on non-convex problems, Material Instabilities in Continuum Mechanics, edited by J.M. Ball, Calderon Press, Oxford, 1988, 269–286.

    Google Scholar 

  111. E.MASCOLO, Local minima of non convex problems, preprint Università di Firenze, 1989.

    Google Scholar 

  112. E. MASCOLO-R. SCHIANCHI, Existence theorems for nonconvex problems, J.Math. Pures Appl., 62 (1983), 349–359.

    MathSciNet  MATH  Google Scholar 

  113. E. MASCOLO-R. SCHIANCHI, Nonconvex problems of the calculus of variations, Nonlinear Anal., 9 (1985), 371–379.

    Article  MathSciNet  Google Scholar 

  114. E. MASCOLO-R. SCHIANCHI, Existence theorems in the calculus of variations, J. Differential Eq., 67 (1987), 185–198.

    Article  MathSciNet  Google Scholar 

  115. B.J. McSHANE, On the necessary condition of Weierstrass in the multiple integral problem of the calculus of variations, Ann. Math., 32 (1931), 578–590.

    Article  MathSciNet  MATH  Google Scholar 

  116. N. MEYERS, Quasiconvexity and lower semicontinuity of multiple integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125–149.

    Article  MathSciNet  MATH  Google Scholar 

  117. M. MIRANDA, Un teorema di esistenza e unicità per il problema dell'area minima in n variabili, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 233–249.

    MathSciNet  MATH  Google Scholar 

  118. C.B. MORREY, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25–53.

    Article  MathSciNet  MATH  Google Scholar 

  119. C.B.MORREY, Multiple integrals in the calculus of variations, D. Grundl. Math. Wiss. 130, Springer-Verlag, 1966.

    Google Scholar 

  120. S. MÜLLER, Weak continuity of determinants and nonlinear elasticity, C.R. Acad. Sc. Paris, 307 (1988), 501–506.

    MathSciNet  MATH  Google Scholar 

  121. F. MURAT, Compacité par compensation, II, Proc. Internat. Meeting on Recent Methods in Nonlinear Analysis, De Giorgi, Magenes, Mosco eds., Pitagora, Bologna, (1979), 245–256.

    Google Scholar 

  122. L. NANIA, On the Γ-convergence for multiple integrals depending on vector valued functions, Ann. Mat. Pura Appl., 134 (1983), 67–77.

    Article  MathSciNet  MATH  Google Scholar 

  123. L. NIRENBERG, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 648–674.

    Article  MathSciNet  MATH  Google Scholar 

  124. R.W. ODGEN, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. R. Soc. London A., 328 (1972), 567–583.

    Article  Google Scholar 

  125. C.OLECH, Integrals of set valued functions and linear optimal control problems, Colloque sur la théorie mathématique du contrôle optimal, C.B.R.M., Vander Louvain, 1970, 109–125.

    Google Scholar 

  126. C. OLECH, A characterization of L1-weak lower semicontinuity of integral functionals, Bull. Polish. Acad. Sci. Math., 25 (1977), 135–142.

    MathSciNet  MATH  Google Scholar 

  127. P. PODIO GIUDUGLI-G. VERGARA CAFFARELLI-E.G. VIRGA, Discontinuous energy minimizers in nonlinear elastostatics; an example of J. Ball revisited, J. Elasticity, 16 (1986), 75–96.

    Article  MathSciNet  MATH  Google Scholar 

  128. J.P. RAYMOND, Champs Hamilioniens, relaxation et existence de solutions en calcul des variations, J. Differential Eq., 70 (1987), 226–274.

    Article  MathSciNet  MATH  Google Scholar 

  129. J.P. RAYMOND, Conditions nécessaires et suffisantes d'existence de solutions en calcul des variations, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 4 (1987), 169–202.

    MathSciNet  MATH  Google Scholar 

  130. J.P. RAYMOND, Théorème d'existence pour des problèmes variationnels non convexes, Proc. Royal Soc. Edinburgh, 107A (1987), 43–64.

    Article  MathSciNet  MATH  Google Scholar 

  131. R.T. ROCKAFELLAR, Convex analysis, Princeton Univ. Press, Princeton, 1970.

    Book  MATH  Google Scholar 

  132. R.T. ROCKAFELLAR, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations, edited by Gossez et al., Lecture Notes in Math. 543, Springer, Berlin, 1975, 157–207.

    Chapter  Google Scholar 

  133. C. SBORDONE, Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa, 2 (1975), 617–638.

    MathSciNet  MATH  Google Scholar 

  134. C.SBORDONE, Lower semicomtinuity and regularity of minima of variational integrals, Nonlinear P.D.E. and Appl., Collège de France Seminar, edited by H.Brézis and J.L.Lions, Pitman, 1983, 194–213.

    Google Scholar 

  135. D. SERRE, Formes quadratiques et calcul des variations, J. Math. Pures Appl., 62 (1983), 117–196.

    MathSciNet  MATH  Google Scholar 

  136. J. SERRIN, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139–167.

    Article  MathSciNet  MATH  Google Scholar 

  137. J. SIVALOGANATHAN, A field theory approach to stability of radial equilibria in nonlinear elasticity, Math. Proc. Cambridge Phil. Soc., 99 (1986), 586–604.

    Article  MathSciNet  MATH  Google Scholar 

  138. J. SIVALOGANATHAN, Implications of rank one convexity, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 5 (1988), 99–118.

    MathSciNet  MATH  Google Scholar 

  139. C.A. STUART, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 2 (1985), 33–66.

    MathSciNet  MATH  Google Scholar 

  140. C.A.STUART, Special problems involving uniqueness and multiplicity in hiperelasticity, preprint.

    Google Scholar 

  141. R. TAHRAOUI, Théorèmes d'existence en calcul des variations et applications à l'élasticité non linéaire, C.R.Acad. Sci. Paris, 302 (1986), 495–498.

    MathSciNet  MATH  Google Scholar 

  142. G.TALENTI, Calcolo delle variazioni, Quaderni dell' Unione Matematica Italiana n. 2, Pitagora Ed., 1977.

    Google Scholar 

  143. L. TARTAR, Compensated compactness, Heriot-Watt Sympos. 4, Pitman, New York, 1978.

    MATH  Google Scholar 

  144. R. TEMAM, A characterization of quasiconvex function, Applied Math. Opt., 8 (1982), 287–291.

    Article  MathSciNet  MATH  Google Scholar 

  145. F.J. TERPSTRA, Die darstellung biquadratischer formen als summen von quadraten mit anwendung auf die variationsrechnung, Math. Ann., 116 (1938), 166–180.

    Article  MathSciNet  MATH  Google Scholar 

  146. L. TONELLI, Fondamenti di calcolo delle variazioni, Volume primo, Zanichelli, Bologna, 1921.

    MATH  Google Scholar 

  147. L. TONELLI, Fondamenti di calcolo delle variazioni, Volume secondo, Zanichelli, Bologna, 1923.

    MATH  Google Scholar 

  148. M. VALADIER, Integration de convexes fermes notamment d'epigraphes in f-convolution continue, R.I.R.O., 4e année, R-2 (1970), 57–73.

    MathSciNet  MATH  Google Scholar 

  149. M. VALADIER, Régularisation s.c.i., relaxation et théorèmes bang-bang, C.R.Acad. Sci. Paris, 293 (1981), 115–116.

    MathSciNet  MATH  Google Scholar 

  150. M. VALADIER, Functions and operators on the space of bounded measures, Atti Sem. Mat. Fis. Univ. Modena, 34 (1985–86), 353–362.

    MathSciNet  MATH  Google Scholar 

  151. L.C. YOUNG, Lectures on the calculus of variations and optimal control theory, W.B.Saunders Company, Philadelphia, 1969.

    MATH  Google Scholar 

  152. V.V. ZHIKOV-A.M. KOZLOV-O.A. OLEINIK-KHA T'EN NGOAN, Averaging and G-convergence of differential operators, Uspeki Math. Nauk, 34 (1979), 65–133; Russian Math. Surveys, 34 (1979), 69–147.

    MathSciNet  MATH  Google Scholar 

  153. T. ZOLEZZI, On equiwellset minimum problems, Appl. Math. Optim., 4 (1978), 209–223.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Paolo Marcellini

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  1. Paolo Marcellini
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Arrigo Cellina

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Marcellini, P. (1990). Non convex integrals of the Calculus of Variations. In: Cellina, A. (eds) Methods of Nonconvex Analysis. Lecture Notes in Mathematics, vol 1446. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084930

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