A theory of integral representation, relaxation and homogenization for some types of variational functionals taking extended real values and possibly being not finite also on large classes of regular functions is presented. Some applications to gradient constrained relaxation and homogenization problems are given.
Sulla rappresentazione integrale, il rilassamento e l'omogeneizzazione di funzionali non limitati. Si presenta una teoria della rappresentazione integrale, rilassamento ed omogeneizzazione per alcune famiglie di funzionali variazionali a valori reali estesi, anche non finiti su ampie classi di funzioni regolari. Si forniscono alcune applicazioni a problemi di rilassamento ed omogeneizzazione con vincoli sul gradiente.
@article{RLIN_1997_9_8_2_129_0,
author = {Luciano Carbone and Riccardo De Arcangelis},
title = {On integral representation, relaxation and homogenization for unbounded functionals},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {8},
year = {1997},
pages = {129-135},
zbl = {0891.49006},
mrnumber = {1485325},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_1997_9_8_2_129_0}
}
Carbone, Luciano; De Arcangelis, Riccardo. On integral representation, relaxation and homogenization for unbounded functionals. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 8 (1997) pp. 129-135. http://gdmltest.u-ga.fr/item/RLIN_1997_9_8_2_129_0/
[1] - , On the relaxation in , of quasi-convex integrals. J. Funct. Anal., 109, 1992, 76-97. | MR 1183605 | Zbl 0769.49009
[2] - - , Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, 5, North-Holland, Amsterdam 1978. | MR 503330 | Zbl 0404.35001
[3] - , Integral representation and relaxation of convex local functional on . Ann. Sc. Norm. Sup. Pisa, 20, 1993, 483-533. | MR 1267597 | Zbl 0802.49008
[ 4 ] - , Equivalence de deux inéquations variationnelles et applications. Arch. Rational Mech. Anal., 41, 1971, 254-265. | MR 346345 | Zbl 0214.11104
[5] , Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Research Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow 1989. | MR 1020296 | Zbl 0669.49005
[6] - , Further results on a problem of homogenization with constraints on the gradient. J. Analyse Math., 44, 1984-85, 1-20. | MR 801284 | Zbl 0574.35006
[7] - , Homogenization of Dirichlet problems with nonnegative bounded constraints on the gradient. J. Analyse Math., 64, 1994, 53-96. | MR 1303508 | Zbl 0822.49012
[8] - , A characterization of families of function sets described by constraints on the gradient. Ann. Inst. Henri Poincaré - Analyse non linéaire, 11, 1994, 553-609. | MR 1302280 | Zbl 0839.49007
[9] , An introduction to -convergence. Progress in Nonlinear Differential Equations and Their Applications, 8, Birkhäuser, Boston-Basel-Berlin1993. | MR 1201152 | Zbl 0816.49001
[10] - , Su un tipo di convergenza variazionale. Atti Acc. Lincei Rend, fis., s. 8, 58, 1975, 842-850. | MR 448194 | Zbl 0339.49005
[11] - , Une notion générale de convergence faible pour des fonctions croissantes d'ensemble. Ann. Sc. Norm. Sup. Pisa, 4, 1977, 61-99. | MR 466479 | Zbl 0405.28008
[12] - , Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-Heidelberg-New York 1976. | MR 521262 | Zbl 0331.35002
[13] - , Convex Analysis and Variational Problems. Studies in Mathematics and its Applications, 1, North-Holland, Amsterdam 1976. | MR 463994 | Zbl 0322.90046
[14] - , Torsion élastoplastique d'une barre cylindrique de section multiconnexe. J. Mécanique, 12, 1973, 151-171. | MR 363114 | Zbl 0273.73023
[15] - , Sublinear functions of measures and variational integrals. Duke Math. J., 31, 1964, 159-178. | MR 162902 | Zbl 0123.09804
[16] - , Electrostatic screening. J. Math. Phys., 16, 1975, 284-288. | MR 382834