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

Corbo Esposito, Antonio; De Arcangelis, Riccardo

Corbo Esposito, Antonio ; De Arcangelis, Riccardo

Annales de l'I.H.P. Analyse non linéaire, Tome 11 (1994) no. 5, pp. 553-609.

EuDML MR Zbl

@article{AIHPC_1994__11_5_553_0,
     author = {Corbo Esposito, Antonio and De Arcangelis, Riccardo},
     title = {A characterization of families of function sets described by constraints on the gradient},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {553--609},
     publisher = {Gauthier-Villars},
     volume = {11},
     number = {5},
     year = {1994},
     mrnumber = {1302280},
     zbl = {0839.49007},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1994__11_5_553_0/}
}

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Corbo Esposito, Antonio; 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, Tome 11 (1994) no. 5, pp. 553-609. http://www.numdam.org/item/AIHPC_1994__11_5_553_0/

Bibliographie
Cité par

[1] E. Acerbi and N. Fusco, Semicontinuity Problems in the Calculus of Variations, Arch. Rational Mech. Anal., 86, 1984, pp. 125-145. | MR | Zbl

[2] H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984. | MR | Zbl

[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. | MR

[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] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978. | MR | Zbl

[6] G. Birkhoff, Integration of Functions with Values in a Banach Space, Trans. Amer. Mat. Soc., 38, 1935, pp. 357-378. | JFM | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[9] G. Bouchitté and M. Valadier, Integral Representation of Convex Functionals on a Space of Measures, J. Funct. Anal., 80, 1988, pp. 398-420. | MR | Zbl

[10] H. Brezis, Multiplicateur de Lagrange en torsion élastoplastique, Arch. Rational Mech. Anal., 49, 1973, pp. 32-40. | MR | Zbl

[11] H. Brezis and M. Sibony, Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal., 41, 1971, pp. 254-265. | MR | Zbl

[12] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France, 96, 1968, pp. 153-180. | Numdam | MR | Zbl

[13] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Longman, Scientific & Technical, 1989. | Zbl

[14] G. Buttazzo and G. Dal Maso, Integral Representation and Relaxation of Local Functionals, Nonlinear Anal., 9, 1985, pp. 515-532. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[18] L. Carbone, Sur un problème d'homogénéisation avec des contraintes sur le gradient, J. Math. Pures Appl., 58, 1979, pp. 275-297. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] L. Carbone and S. Salerno, Homogenization with Unbounded Constraints on the Gradient, Nonlinear Anal., 9, 1985, pp. 431-444. | MR | Zbl

[22] L. Carbone and C. Sbordone, Some Properties of Γ-limits of Integral Functionals, Ann. Math. Pura Appl., 122, 1979, pp. 1-60. | MR | Zbl

[23] D. Cioranescu and J. Saint Jean Paulin, Homogenization in Open Sets with Holes, J. Math. Pures Appl., 71, 1979, pp. 590-607. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[26] A. Corbo Esposito and R. De Arcangelis, Homogenization of Dirichlet Problems with Nonnegative Bounded Constraints on the Gradient, to appear onJ. Analyse Math. | Zbl

[27] G. Dal Maso, Integral Representation on BV (Ω) of Γ-limits of Variational Integrals, Manuscripta Math., 30, 1980, pp. 387-413. | MR | Zbl

[28] G. Dal Maso and L. Modica, A General Theory of Variational Functionals, Topics in Functional Analysis 1980/81, Quaderni della Scuola Normale Superiore, Pisa, 1982, pp 149-221. | MR | Zbl

[29] R. De Arcangelis, A General Homogenization Result for Almost Periodic Functionals, J. Math. Anal. Appl., 156, 1991, pp. 358-380. | MR | Zbl

[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. | MR | Zbl

[3 1 ] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rend. Mat. Roma, 8, 1975, pp. 277-294. | MR | Zbl

[32] E. Di Giorgi, Convergence Problems for Functionals and Operators, Proceed. Int. Meeting on "Recent Methods in Nonlinear Analysis", Rome 8-12 May 1975, Pitagora ed., Bologna, 1979, pp. 131-188. | MR | Zbl

[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. | MR | Zbl

[34] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3, 1979, pp. 63-101.

[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. | Numdam | MR | Zbl

[36] F. Demengel and R. Temam, Convex Functions of a Measure and Applications, Indiana Univ. Math. J., 33, 5, 1984, pp. 673-709. | MR | Zbl

[37] N. Dunford and J.T. Schwartz, Linear Operators, Wiley-Interscience Publications, 1957. | Zbl

[38] G. Duvaut and J.-L. Lions, Les Inéquations en mécanique et en physique, Dunod, Paris, 1972. | MR | Zbl

[39] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland American Elsevier, 1976. | MR | Zbl

[40] N. Fusco, Γ-convergenza unidimensionale, Boll. Un. Mat. Ital. B , 16-B, 1979, pp. 74-86. | MR | Zbl

[41] E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser-Verlag, Basel, 1984. | MR | Zbl

[42] R. Glowinski and H. Lanchon, Torsion élastoplastique d'une barre cylindrique de section multiconnexe, J. Mech., 12, 1973, pp. 151-171. | MR | Zbl

[43] C. Goffman and J. Serrin, Sublinear Functions of Measures and Variational Integrals, Duke Math. J., 31, 1964, pp. 159-178. | MR | Zbl

[44] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, Pure and Applied Mathematics, 1980. | MR | Zbl

[45] K. Kuratowski, Topology, Academic Press, New York, 1966. | MR | Zbl

[46] H. Lanchon, Torsion élastoplastique d'une barre cylindrique de section simplement ou multiplement connexe, J. Mech., 13, 1974, pp. 267-320. | MR | Zbl

[47] P. Marcellini and C. Sbordone, Homogenization of non Uniformly Elliptic Operators, Applicable Analysis, 8, 1978, pp. 101-113. | MR | Zbl

[48] R.S. Phillips, Integration in a Convex Linear Topological Space, Trans. Amer. Mat. Soc., 47, 1940, pp. 114-145. | MR | Zbl

[49] W. Rudin, Analisi Reale e Complessa, Boringhieri, Torino, 1980.

[50] C. Sbordone, Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Sc. Norm. Sup. Pisa, 2, 1975, pp. 617-638. | Numdam | MR | Zbl

[51] J. Serrin, On the Definition and Properties of Certain Variational Integrals, Trans. Amer. Math. Soc., 101, 1961, pp. 139-167. | MR | Zbl

[52] S. Spagnolo, Sulla convergenza delle soluzioni di equazioni paraboliche ed ellittiche, Ann. Sc. Norm. Sup. Pisa, 22, 1968, pp. 577-597. | Numdam | MR | Zbl

[53] T.W. Ting, Elastic-plastic Torsion, Arch. Rational Mech. Anal, 24, 1969, pp. 228-244. | MR | Zbl