A characterization of families of function sets described by constraints on the gradient

Antonio Corbo Esposito; Riccardo De Arcangelis

Annales de l'I.H.P. Analyse non linéaire (1994)

  • Volume: 11, Issue: 5, page 553-609
  • ISSN: 0294-1449

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Corbo Esposito, Antonio, and De Arcangelis, Riccardo. "A characterization of families of function sets described by constraints on the gradient." Annales de l'I.H.P. Analyse non linéaire 11.5 (1994): 553-609. <http://eudml.org/doc/78344>.

@article{CorboEsposito1994,
author = {Corbo Esposito, Antonio, De Arcangelis, Riccardo},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {constraints; gradient; Sobolev spaces; families of subsets; homogenization; elastic-plastic torsion},
language = {eng},
number = {5},
pages = {553-609},
publisher = {Gauthier-Villars},
title = {A characterization of families of function sets described by constraints on the gradient},
url = {http://eudml.org/doc/78344},
volume = {11},
year = {1994},
}

TY - JOUR
AU - Corbo Esposito, Antonio
AU - De Arcangelis, Riccardo
TI - A characterization of families of function sets described by constraints on the gradient
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1994
PB - Gauthier-Villars
VL - 11
IS - 5
SP - 553
EP - 609
LA - eng
KW - constraints; gradient; Sobolev spaces; families of subsets; homogenization; elastic-plastic torsion
UR - http://eudml.org/doc/78344
ER -

References

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  1. [1] E. Acerbi and N. Fusco, Semicontinuity Problems in the Calculus of Variations, Arch. Rational Mech. Anal., 86, 1984, pp. 125-145. Zbl0565.49010MR751305
  2. [2] H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984. Zbl0561.49012MR773850
  3. [3] H. Attouch and C. Picard, Variational Inequalities with Varying Obstacles: The General Form of the Limit Problem, J. Funct. Anal., 50, 1983, pp. 329-386. MR695419
  4. [4] C. Baiocchi and A. Capelo, Disequazioni variazionali e quasivariazionali. Applicazioni a problemi di frontiera libera. Vol. 1: Problemi variazionali, Quaderni Unione Matematica Italiana, Pitagora Editrice, Bologna, 1978. 
  5. [5] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978. Zbl0404.35001MR503330
  6. [6] G. Birkhoff, Integration of Functions with Values in a Banach Space, Trans. Amer. Mat. Soc., 38, 1935, pp. 357-378. Zbl0013.00803MR1501815JFM61.0234.01
  7. [7] G. Bouchitté, Convergence et relaxation de fonctionnelles du calcul des variations à croissance linéaire. Applications à l'homogénéisation en plasticité, Ann. Fac. Sci. Univ. Toulouse, Math., VIII, 1, 1986/87, pp. 7-36. Zbl0663.49002MR899595
  8. [8] G. Bouchitté and G. Dal Maso, Integral Representation and Relaxation of Convex Local Functionals on BV (Ω), Ann. Sc. Norm. Sup. Pisa, IV, 20, 1993, pp. 483-533. Zbl0802.49008MR1267597
  9. [9] G. Bouchitté and M. Valadier, Integral Representation of Convex Functionals on a Space of Measures, J. Funct. Anal., 80, 1988, pp. 398-420. Zbl0662.46009MR961907
  10. [10] H. Brezis, Multiplicateur de Lagrange en torsion élastoplastique, Arch. Rational Mech. Anal., 49, 1973, pp. 32-40. Zbl0265.35021MR346346
  11. [11] H. Brezis and M. Sibony, Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal., 41, 1971, pp. 254-265. Zbl0214.11104MR346345
  12. [12] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France, 96, 1968, pp. 153-180. Zbl0165.45601MR239302
  13. [13] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in theCalculus of Variations, Longman, Scientific & Technical, 1989. Zbl0669.49005
  14. [14] G. Buttazzo and G. Dal Maso, Integral Representation and Relaxation of Local Functionals, Nonlinear Anal., 9, 1985, pp. 515-532. Zbl0527.49008MR794824
  15. [15] L. Carbone, Sur la convergence des intégrales du type de l'énergie sur des fonctions à gradient borné, J. Math. Pures Appl., 56, 9, 1977, pp. 79-84. Zbl0308.49023MR482459
  16. [16] L. Carbone, Γ-convergence d'intégrales sur des fonctions avec des contraintes sur le gradient, Comm. Part. Diff. Eq., 2, 1977, pp. 627-651. Zbl0357.49020MR493642
  17. [17] L. Carbone, Sull' omogeneizzazione di un problema variazionale con vincoli sul gradiente, Atti Accad. Naz. Lincei, Rend. Cl.Sci. Fis. Mat. Natur., 63, 8, 1977, pp. 10-14. Zbl0395.49029
  18. [18] L. Carbone, Sur un problème d'homogénéisation avec des contraintes sur le gradient, J. Math. Pures Appl., 58, 1979, pp. 275-297. Zbl0387.49028MR544254
  19. [19] L. Carbone and S. Salerno, On a Problem of Homogenization with Quickly Oscillating Constraints on the Gradient, J. Math. Anal. Appl., 90, 1982, pp. 219-250. Zbl0499.49007MR680876
  20. [20] L. Carbone and S. Salerno, Further Results on a Problem of Homogenization with Constraints on the Gradient, J. Analyse Math. , 44, 1984/85, pp. 1-20. Zbl0574.35006MR801284
  21. [21] L. Carbone and S. Salerno, Homogenization with Unbounded Constraints on the Gradient, Nonlinear Anal., 9, 1985, pp. 431-444. Zbl0557.49006MR785715
  22. [22] L. Carbone and C. Sbordone, Some Properties of Γ-limits of Integral Functionals, Ann. Math. Pura Appl., 122, 1979, pp. 1-60. Zbl0474.49016MR565062
  23. [23] D. Cioranescu and J. Saint Jean Paulin, Homogenization in Open Sets with Holes, J. Math. Pures Appl., 71, 1979, pp. 590-607. Zbl0427.35073MR548785
  24. [24] A. Corbo Esposito and R. De Arcangelis, Comparison Results for Some Types of Relaxation of Variational Integral Functionals, Ann. Mat. Pura Appl., 164, 1993, pp. 155-193. Zbl0931.49009MR1243954
  25. [25] A. Corbo Esposito and R. De Arcangelis, The Lavrentieff Phenomenon and Different Processes of Homogenization, Comm. Part. Diff. Eq., 17, 1992, pp. 1503-1538. Zbl0814.35006MR1187620
  26. [26] A. Corbo Esposito and R. De Arcangelis, Homogenization of Dirichlet Problems with Nonnegative Bounded Constraints on the Gradient, to appear onJ. Analyse Math. Zbl0822.49012
  27. [27] G. Dal Maso, Integral Representation on BV (Ω) of Γ-limits of Variational Integrals, Manuscripta Math., 30, 1980, pp. 387-413. Zbl0435.49016MR567216
  28. [28] G. Dal Maso and L. Modica, A General Theory of Variational Functionals, Topics in Functional Analysis1980/81, Quaderni della Scuola Normale Superiore, Pisa, 1982, pp 149-221. Zbl0493.49005MR671757
  29. [29] R. De Arcangelis, A General Homogenization Result for Almost Periodic Functionals, J. Math. Anal. Appl., 156, 1991, pp. 358-380. Zbl0743.49019MR1103018
  30. [30] R. De Arcangelis and A. Vitolo, Some Cases of Homogenization with Unbounded Oscillating Constraints on the Gradient, Asymp. Anal., 5, 1992, pp. 397-428. Zbl0769.49029MR1159380
  31. [3 1 ] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rend. Mat. Roma, 8, 1975, pp. 277-294. Zbl0316.35036MR375037
  32. [32] E. Di Giorgi, Convergence Problems for Functionals and Operators, Proceed. Int. Meeting on "Recent Methods in Nonlinear Analysis", Rome8-12 May 1975, Pitagora ed., Bologna, 1979, pp. 131-188. Zbl0405.49001MR533166
  33. [33] E. De Giorgi, F. Colombini and L.C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Quaderni della Scuola Normale Superiore, Pisa, 1972. Zbl0296.49031MR493669
  34. [34] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3, 1979, pp. 63-101. 
  35. [35] E. De Giorgi and G. Letta, Une notion générale de convergence faible pour des fonctions croissantes d'ensemble, Ann. Sc. Norm. Sup. Pisa, (4), 4, 1977, pp. 61-99. Zbl0405.28008MR466479
  36. [36] F. Demengel and R. Temam, Convex Functions of a Measure and Applications, Indiana Univ. Math. J., 33, 5, 1984, pp. 673-709. Zbl0581.46036MR756154
  37. [37] N. Dunford and J.T. Schwartz, Linear Operators, Wiley-Interscience Publications, 1957. Zbl0084.10402
  38. [38] G. Duvaut and J.-L. Lions, Les Inéquations en mécanique et en physique, Dunod, Paris, 1972. Zbl0298.73001MR464857
  39. [39] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland American Elsevier, 1976. Zbl0322.90046MR463994
  40. [40] N. Fusco, Γ-convergenza unidimensionale, Boll. Un. Mat. Ital. B , 16–B, 1979, pp. 74-86. Zbl0403.49015MR536528
  41. [41] E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser-Verlag, Basel, 1984. Zbl0545.49018MR775682
  42. [42] R. Glowinski and H. Lanchon, Torsion élastoplastique d'une barre cylindrique de section multiconnexe, J. Mech., 12, 1973, pp. 151-171. Zbl0273.73023MR363114
  43. [43] C. Goffman and J. Serrin, Sublinear Functions of Measures and Variational Integrals, Duke Math. J., 31, 1964, pp. 159-178. Zbl0123.09804MR162902
  44. [44] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, Pure and Applied Mathematics, 1980. Zbl0457.35001MR567696
  45. [45] K. Kuratowski, Topology, Academic Press, New York, 1966. Zbl0158.40802MR217751
  46. [46] H. Lanchon, Torsion élastoplastique d'une barre cylindrique de section simplement ou multiplement connexe, J. Mech., 13, 1974, pp. 267-320. Zbl0285.73020MR363107
  47. [47] P. Marcellini and C. Sbordone, Homogenization of non Uniformly Elliptic Operators, Applicable Analysis, 8, 1978, pp. 101-113. Zbl0406.35014MR523948
  48. [48] R.S. Phillips, Integration in a Convex Linear Topological Space, Trans. Amer. Mat. Soc., 47, 1940, pp. 114-145. Zbl0022.31902MR2707
  49. [49] W. Rudin, Analisi Reale e Complessa, Boringhieri, Torino, 1980. 
  50. [50] C. Sbordone, Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Sc. Norm. Sup. Pisa, 2, 1975, pp. 617-638. Zbl0317.49012MR417753
  51. [51] J. Serrin, On the Definition and Properties of Certain Variational Integrals, Trans. Amer. Math. Soc., 101, 1961, pp. 139-167. Zbl0102.04601MR138018
  52. [52] S. Spagnolo, Sulla convergenza delle soluzioni di equazioni paraboliche ed ellittiche, Ann. Sc. Norm. Sup. Pisa, 22, 1968, pp. 577-597. Zbl0174.42101MR240443
  53. [53] T.W. Ting, Elastic-plastic Torsion, Arch. Rational Mech. Anal, 24, 1969, pp. 228-244. Zbl0179.53903MR264889

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