Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 53-83.

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses can be assumed, and in which the results can be obtained in the general framework of BV spaces. The classical homogenization result is established for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained, and with an explicitly computed homogeneous density.

DOI : 10.1051/cocv:2003032
Classification : 49J45, 49N20, 74Q05
Mots-clés : homogenization, gradient constrained variational problems, nonlinear elastomers
Carbone, Luciano  ; Cioranescu, Doina  ; Arcangelis, Riccardo De  ; Gaudiello, Antonio 1

1 Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;
@article{COCV_2004__10_1_53_0,
     author = {Carbone, Luciano and Cioranescu, Doina and Arcangelis, Riccardo De and Gaudiello, Antonio},
     title = {Homogenization of unbounded functionals and nonlinear elastomers. {The} case of the fixed constraints set},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {53--83},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {1},
     year = {2004},
     doi = {10.1051/cocv:2003032},
     mrnumber = {2084255},
     zbl = {1072.49008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003032/}
}
TY  - JOUR
AU  - Carbone, Luciano
AU  - Cioranescu, Doina
AU  - Arcangelis, Riccardo De
AU  - Gaudiello, Antonio
TI  - Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
SP  - 53
EP  - 83
VL  - 10
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003032/
DO  - 10.1051/cocv:2003032
LA  - en
ID  - COCV_2004__10_1_53_0
ER  - 
%0 Journal Article
%A Carbone, Luciano
%A Cioranescu, Doina
%A Arcangelis, Riccardo De
%A Gaudiello, Antonio
%T Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 53-83
%V 10
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2003032/
%R 10.1051/cocv:2003032
%G en
%F COCV_2004__10_1_53_0
Carbone, Luciano; Cioranescu, Doina; Arcangelis, Riccardo De; Gaudiello, Antonio. Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 53-83. doi : 10.1051/cocv:2003032. http://www.numdam.org/articles/10.1051/cocv:2003032/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Claredon Press, Oxford Math. Monogr. (2000). | MR | Zbl

[2] A.H.T. Banks, N.J. Lybeck, B. Munoz and L. Yanyo, Nonlinear Elastomers: Modeling and Estimation, in Proc. of the “Third IEEE Mediterranean Symposium on New Directions in Control and Automation”, Vol. 1. Limassol, Cyprus (1995) 1-7.

[3] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North Holland, Stud. Math. Appl. 5 (1978). | MR | Zbl

[4] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford Lecture Ser. Math. Appl. 12 (1998). | MR | Zbl

[5] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Longman Scientific & Technical, Pitman Res. Notes Math. Ser. 207 (1989). | MR | Zbl

[6] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, An Approach to the Homogenization of Nonlinear Elastomers via the Theory of Unbounded Functionals. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 283-288. | MR | Zbl

[7] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, Homogenization of Unbounded Functionals and Nonlinear Elastomers. The General Case. Asymptot. Anal. 29 (2002) 221-272. | MR | Zbl

[8] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, An Approach to the Homogenization of Nonlinear Elastomers in the Case of the Fixed Constraints Set. Rend. Accad. Sci. Fis. Mat. Napoli (4) 67 (2000) 235-244. | MR

[9] L. Carbone and R. De Arcangelis, On Integral Representation, Relaxation and Homogenization for Unbounded Functionals. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 (1997) 129-135. | MR | Zbl

[10] L. Carbone and R. De Arcangelis, On the Relaxation of Some Classes of Unbounded Integral Functionals. Matematiche 51 (1996) 221-256; Special Issue in honor of Francesco Guglielmino. | MR | Zbl

[11] L. Carbone and R. De Arcangelis, Unbounded Functionals: Applications to the Homogenization of Gradient Constrained Problems. Ricerche Mat. 48-Suppl. (1999) 139-182. | MR | Zbl

[12] L. Carbone and R. De Arcangelis, On the Relaxation of Dirichlet Minimum Problems for Some Classes of Unbounded Integral Functionals. Ricerche Mat. 48 (1999) 347-372; Special Issue in memory of Ennio De Giorgi. | MR | Zbl

[13] L. Carbone and R. De Arcangelis, On the Unique Extension Problem for Functionals of the Calculus of Variations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001) 85-106. | MR

[14] L. Carbone and S. Salerno, Further Results on a Problem of Homogenization with Constraints on the Gradient. J. Analyse Math. 44 (1984/85) 1-20. | MR | Zbl

[15] D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press, Oxford Lecture Ser. Math. Appl. 17 (1999). | MR | Zbl

[16] A. Corbo Esposito and R. De Arcangelis, The Lavrentieff Phenomenon and Different Processes of Homogenization. Comm. Partial Differential Equations 17 (1992) 1503-1538. | MR | Zbl

[17] A. Corbo Esposito and R. De Arcangelis, Homogenization of Dirichlet Problems with Nonnegative Bounded Constraints on the Gradient. J. Analyse Math. 64 (1994) 53-96. | MR | Zbl

[18] A. Corbo Esposito and F. Serra Cassano, A Lavrentieff Phenomenon for Problems of Homogenization with Constraints on the Gradient. Ricerche Mat. 46 (1997) 127-159. | MR | Zbl

[19] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser-Verlag, Progr. Nonlinear Differential Equations Appl. 8 (1993). | MR | Zbl

[20] C. D'Apice, T. Durante and A. Gaudiello, Some New Results on a Lavrentieff Phenomenon for Problems of Homogenization with Constraints on the Gradient. Matematiche 54 (1999) 3-47. | Zbl

[21] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842-850. | MR | Zbl

[22] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Grundlehren Math. Wiss. 219 (1976). | MR | Zbl

[23] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton Math. Ser. 28 (1972). | MR | Zbl

[24] L.R.G. Treloar, The Physics of Rubber Elasticity. Clarendon Press, Oxford, First Ed. (1949), Third Ed. (1975).

[25] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Grad. Texts in Math. 120 (1989). | MR | Zbl

Cité par Sources :