Homogenization of a quasi-linear problem with quadratic growth in perforated domains : an example

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1. [1] G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rat. Mech. Anal., Vol. 133, 1, 1995, pp. 77-101. Zbl0859.35031MR1367357
2. [2] A. Bensoussan, L. Boccardo and F. Murat, H-Convergence for quasilinear elliptic equations with quadratic growth, Appl. Math. Optim., Vol. 26, 3, 1992, pp. 253-272. Zbl0795.35008MR1175481
3. [3] A. Bensoussan, L. Boccardo, A. Dall'aglio and F. Murat, H-convergence for quasilinear elliptic equations under natural hypotheses on the correctors. In Composite Media and Homogenization Theory II (Proceedings, Trieste, September 1993), ed. by G. DAL MASO, G. F. DELL'ANTONIO, 1995, World Scientific, Singapore, pp. 93-112. 
4. [4] L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique. In Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. IV, ed. by H. BRÉZIS, J.-L. LIONS. Research Notes in Math., Vol. 84, 1983, Pitman, London, pp. 19-73. Zbl0588.35041MR716511
5. [5] J. Casado-Díaz, Sobre la homogeneización de problemas no coercivos y problemas en dominios con agujeros. Ph. D. Thesis, University of Seville, 1993. 
6. [6] J. Casado-Díaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains. To appear in J. Math. Pur. App. Zbl0880.35015MR1460666
7. [7] J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains. To appear in P. Roy. Soc. Edimburgh. Zbl0877.35013MR1453278
8. [8] J. Casado-Díaz and A. Garroni, Asymptotic behaviour of Dirichlet solutions of nonlinear elliptic systems in varying domains. To appear. Zbl0952.35036
9. [9] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs. In Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. II and III, ed. by H. BRÉZIS, J.-L. LIONS. Research Notes in Math. Vol. 60 and 70, 1982, Pitman, London, pp. 98-138 and pp. 154-178. Zbl0496.35030
10. [10] G. Dal Maso and A. Garroni, New results on the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Mod. Meth. Appl. Sci., Vol. 3, 1994, pp. 373-407. Zbl0804.47050MR1282241
11. [11] G. Dal Maso and U. Mosco, Wiener criterion and Γ — convergence. Appl. Math. Optim. Vol. 151987, pp. 15-63. Zbl0644.35033MR866165
12. [12] G. Dal Maso and U. Mosco, Wiener-criteria and energy decay for relaxed Dirichlet problems. Arch. Rat. Mech. Anal., Vol. 95, 4, 1986, pp. 345-387. Zbl0634.35033MR853783
13. [13] G. Dal Maso and F. Murat, Dirichlet problems in perforated domains for homogeneous monotone operators on H10. In Calculus of Variations, Homogenization and Continuum Mechanics (Proceedings, Cirm-Lumigny, Marseille, June 21-25, 1993), ed. by G. Bouchitté, G. Buttazzo, P. Suquet. Series on Advances in Mathematics for Applied Sciences18, World Scientific, Singapore, 1994, pp. 177-202. Zbl0884.47026MR1428699
14. [14] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. To appear. Zbl0899.35007MR1487956
15. [15] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics., CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
16. [16] H. Federer and W. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana U. Math. J., Vol. 22, 1972, 139-158. Zbl0238.28015MR435361
17. [17] S. Finzi Vita, F. Murat and N.A. Tchou, Quasilinear relaxed Dirichlet problems. Siam J. Math. Anal., Vol. 27, 4, 1996, pp. 977-996. Zbl0863.35041MR1393419
18. [18] S. Finzi Vita and N.A. Tchou, Corrector results for relaxed Dirichlet problems. Asymptotic Analysis, Vol. 5, 1992, pp. 269-281. Zbl0784.35020MR1145113
19. [19] H. Kacimi and F. Murat, Estimation de l'erreur dans des problèmes de Dirichlet où apparaît un terme étrange. In Partial Differential Equations and the Calculus of Variations, Vol. II, Essays in honor of E. De Giorgi, ed. by F. Colombini, L. Modica, A. Marino, S. Spagnolo. Progress in Nonlinear Differential Equations and their Applications2, Birkhäuser, Boston, 1989, pp. 661-696. Zbl0687.35007MR1034024
20. [20] N.G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa C. Sci., Vol. 3, 17, 1963, pp. 189-206. Zbl0127.31904MR159110
21. [21] F. Murat, H-convergence.Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger1977/1978, mimeographed notes. English translation: F. MURAT, L. TARTAR, H-convergence. In Mathematical Modelling of Composites Materials, ed. by R. V. KOHN. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston. To appear. 
22. [22] F. Murat, Homogenization of renormalized solutions of elliptic equations. Ann. Inst. H. Poincaré, Anal. non linéaire, Vol. 8, 1991, pp. 309-332. Zbl0774.35007MR1127929
23. [23] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, 1989. Zbl0692.46022MR1014685

1. Gianni Dal Maso, François Murat, Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators