, O(x, y) is an admissible test function in the sense of Definition 5.1, i.e., satisfies (5.1) limla-o h(x,)dx=Ia IY Ib(x,y)ldxdy

, Before proving Lemma 5.5, let us remark that any function satisfying (i)-(iii) obviously belongs to

, The converse is more subtle. Indeed, since b(x, y) is an element of C[; L(Y)], for each x eft, its value y-q(x, y) is a class of functions in L( Y)" picking up a representative for each x and collecting them gives a "representative

, LEMMA 5.6. Let d/(x,y) be a function in C[f

. Then, there exists a "representative" of (x, y)for which properties (i)-(iii) in Lemma 5.5 hold

, By definition, for any value of x eft, the function y-O(x, y) is measurable on Y, Y-periodic, and there exists a subset E(x) of measure zero in Y such that 0(x,y) is bounded on Y-E(x). The continuity of x-q(x, y) from f in L(Y) is equivalent to (5.10) lim Sup

Y. ,

, We emphasize that, a priori, the exceptional set E(x), where the function y 4(x, y) is not defined, depends on x. Nevertheless, thanks to the continuity of b(x, y) with respect to the x variable

, Let (K)__ be a sequence of partitions of K (i.e., U 7= K K and ]K, K21 0 if j) such that lim._,+o Sup_i. diam (K)=0. Let X(x) be the characteristic function of K, and x a point in K. Define the step function, vol.4

E. Acerbi, V. Chiado, G. Piat, . Dal, A. D. Maso et al., An extension theorem from connected sets, and homogenization in general periodic domains

G. Allaire, Homogdndisation et convergence fi deux dchelles, application gt un problme de convection diffusion, C. R. Acad. Sci, pp.581-586, 1991.

, Homogenization of the unsteady Stokes equations in porous media, Proceedings of the 1st

, European Conference on Elliptic and Parabolic Problems, 1991.

G. A. Allaire and . Murat, Homogenization of the Neuman problem with non-isolated holes, Asymptotic Anal

Y. Amirat, K. Hamdache, and A. A. Ziani, Homogdndisation non-locale pour des dquations ddgdndrdes fi codfficients pdriodiques, C. R. Acad. Sci, pp.963-966, 1991.

Z. Arbogast, J. Douglas, and A. U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal, vol.21, pp.823-836, 1990.

N. Bakhvalov-and-g and . Panasenko, Homogenization: averaging processes in periodic media, Math. Appl, vol.36, 1990.

J. Ball, A version of the fundamental theorem for Young measures, in Pde's and continuum models of phase transitions, Lecture Notes in Phys, vol.344, 1989.

J. Ball-and-f and . Murat, Wl'P-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal, vol.58, pp.225-253, 1984.

A. Bensoussan, J. L. Lions, and A. G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 1978.

N. Bourbaki, Eldments de Mathdmatiques, 1965.

A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL, vol.103, pp.313-322, 1985.

D. Paulin, Homogenization in Open Sets with Holes, J. Math. Anal. Appl, vol.71, pp.590-607, 1979.

M. Crandall, H. Ishii, and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc

G. Dal-maso-and-a and . Defranceschi, Correctors for the homogenization of monotone operators, Differential Integral Equations, vol.3, pp.1151-1166, 1990.

E. De-giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rend. Mat, vol.8, pp.277-294, 1975.

G. and F. , Proceedings of the International Congress of Mathematicians Warsazwa, pp.1175-1191, 1983.

R. Diperna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal, vol.88, pp.223-270, 1985.

W. E. , Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math, vol.45, pp.301-326, 1992.

A. Grtgoire,

W. E. Serre, Correctors for the homogenization of conservation laws with oscillatory forcing terms, Asymptotic Anal, vol.5, pp.311-316, 1992.

I. Ekeland-and-r and . Temam, Analyse convexe et problmes variationnels, 1974.

L. C. Evans, The perturbed test function method for viscosity solutions of non-linear partial differential equations, Proc. Roy. Soc. Edinburgh

, Periodic homogenization of certain fully non-linear partial differential equations

G. Francfort,

P. A. Glrard and . Murat,

T. A. Hou and . Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math, vol.52, pp.34-45, 1992.

J. B. Keller, Darcy's law for flow in porous media and the two-space method, Lecture Notes in Pure and Appl. Math, vol.54, 1980.

S. Kozlov, O. Oleinik, and A. V. Zhikov, Homogenization ofparabolic operators with almost-periodic coefficients, Mat. Sbornik, vol.117, pp.69-85, 1982.

J. L. Lions, Proceedings Internat. Meeting on Recent Methods in Non-Linear Analysis, pp.189-203, 1979.

, Some Methods in the Mathematical Analysis of Systems and Their Control, 1981.

P. Marcellini, Periodic solutions and homogenization of non linear variational problems, Ann. Mat. Pura Appl, vol.117, issue.4, pp.139-152, 1978.

L. Mascarenhas and F. -limite-d, unefonctionnelle lide ?tun phdnom.ne de mdmoire, C. R. Acad. Sci. Paris, pp.67-70, 1991.

S. Muller, Homogenization of nonconvex integral functionals and cellular materials, Arch. Rational Mech. Anal, pp.189-212, 1988.

F. Murat and H. -convergence, Analyse Fonctionnelle et Num6rique de l'Universit6 d'Alger, mimeographed notes, 1978.

, Correctors for monotone problems in non-periodic homogenization

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal, vol.20, pp.608-623, 1989.

, Asymptotic analysis for a stiff variational problem arising in mechanics, SIAM J. Math. Anal, vol.21, pp.1394-1414, 1990.

O. Oleinik-and-v and . Zhikov, On the homogenization ofelliptic operators with almost-periodic coefficients, Rend. Sem. Mat. Fis. Milano, vol.52, pp.149-166, 1982.

G. Panasenko, Multicomponent homogenization ofprocesses in strongly nonhomogeneous structures, Math. USSR Sbornik, vol.69, pp.143-153, 1991.

E. Sanchez-palencia, Nonhomogeneous media and vibration theory, vol.127, 1980.

S. Spagnolo, Convergence in energy for elliptic operators, Numerical Solutions of Partial Differential Equations III Synspade, 1975.

L. Tartar, C. Peccot-au-collge-de, and F. ,

, Topics in Nonlinear Analysis, Publications math6matiques d'Orsay 78, vol.13, 1978.

, Convergence of the homogenization process, Appendix of Nonhomogeneous media and vibration theory, Lecture Notes in Phys, vol.127, 1980.

, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics, Heriot-Watt Symposium IV, Research Notes in Math, pp.136-212, 1979.

, Nonlocal effects induced by homogenization, Partial Differential Equations and the Calculus of Variations, 1989.

R. Temam, Navier-Stokes Equations, 1979.

K. Yosida, Functional Analysis, 1964.

V. Zhikov, S. Kozlov, O. Oleinik, and A. K. Ngoan, Averaging and G-convergence of differential operators, Russian Math. Surveys, vol.34, pp.69-147, 1979.