, 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
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
,
, 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
An extension theorem from connected sets, and homogenization in general periodic domains ,
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.
Homogenization of the Neuman problem with non-isolated holes, Asymptotic Anal ,
Homogdndisation non-locale pour des dquations ddgdndrdes fi codfficients pdriodiques, C. R. Acad. Sci, pp.963-966, 1991. ,
Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal, vol.21, pp.823-836, 1990. ,
Homogenization: averaging processes in periodic media, Math. Appl, vol.36, 1990. ,
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. ,
Wl'P-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal, vol.58, pp.225-253, 1984. ,
Asymptotic Analysis for Periodic Structures, 1978. ,
, Eldments de Mathdmatiques, 1965.
Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL, vol.103, pp.313-322, 1985. ,
Homogenization in Open Sets with Holes, J. Math. Anal. Appl, vol.71, pp.590-607, 1979. ,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc ,
Correctors for the homogenization of monotone operators, Differential Integral Equations, vol.3, pp.1151-1166, 1990. ,
Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rend. Mat, vol.8, pp.277-294, 1975. ,
, Proceedings of the International Congress of Mathematicians Warsazwa, pp.1175-1191, 1983.
Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal, vol.88, pp.223-270, 1985. ,
Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math, vol.45, pp.301-326, 1992. ,
,
Correctors for the homogenization of conservation laws with oscillatory forcing terms, Asymptotic Anal, vol.5, pp.311-316, 1992. ,
, Analyse convexe et problmes variationnels, 1974.
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
,
,
Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl. Math, vol.52, pp.34-45, 1992. ,
Darcy's law for flow in porous media and the two-space method, Lecture Notes in Pure and Appl. Math, vol.54, 1980. ,
Homogenization ofparabolic operators with almost-periodic coefficients, Mat. Sbornik, vol.117, pp.69-85, 1982. ,
, 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.
Periodic solutions and homogenization of non linear variational problems, Ann. Mat. Pura Appl, vol.117, issue.4, pp.139-152, 1978. ,
unefonctionnelle lide ?tun phdnom.ne de mdmoire, C. R. Acad. Sci. Paris, pp.67-70, 1991. ,
Homogenization of nonconvex integral functionals and cellular materials, Arch. Rational Mech. Anal, pp.189-212, 1988. ,
Analyse Fonctionnelle et Num6rique de l'Universit6 d'Alger, mimeographed notes, 1978. ,
, Correctors for monotone problems in non-periodic homogenization
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.
On the homogenization ofelliptic operators with almost-periodic coefficients, Rend. Sem. Mat. Fis. Milano, vol.52, pp.149-166, 1982. ,
Multicomponent homogenization ofprocesses in strongly nonhomogeneous structures, Math. USSR Sbornik, vol.69, pp.143-153, 1991. ,
, Nonhomogeneous media and vibration theory, vol.127, 1980.
Convergence in energy for elliptic operators, Numerical Solutions of Partial Differential Equations III Synspade, 1975. ,
,
, 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.
, Navier-Stokes Equations, 1979.
, Functional Analysis, 1964.
Averaging and G-convergence of differential operators, Russian Math. Surveys, vol.34, pp.69-147, 1979. ,