http://cmaf.ptmat.fc.ul.pt/prepubmat/ Pre-print UL-MAT 2004-002 Bounds for non periodic mixtures of infinitely many materials Cristian Barbarosie and Anca-Maria Toader CMAF, Faculty of Sciences, University of Lisbon Corresponding author: Cristian Barbarosie CMAF, Av. Prof. Gama Pinto, 2 1649-003 Lisboa, Portugal tel. + 351 217904902, fax + 351 217954288 e-mail: barbaros@ptmat.fc.ul.pt Summary We prove bounds on the homogenized coefficients for general non periodic mixtures of an arbitrary number of isotropic materials, in the heat conduction framework. The component materials and their proportions are given through the Young measure associated to the sequence of coefficient functions. Upper and lower bounds inequalities are deduced in terms of algebraic relations between this Young measure and the eigenvalues of the H -limit matrix. The proofs employ arguments of compensated compactness and fine properties of Young measures. When restricted to the periodic case, we recover known bounds. AMS Subject Classification: 35B27, 73B27, 49K20, 49N45. Keywords: homogenization, composite materials, optimal bounds, Young measures. 1. State of the art Homogenization theory has developed following two approaches: the general case (under no additional hypothesis on the material coefficients) and the periodic case (assuming pe- riodicity of the coefficients). The periodicity assumption is restrictive from the theoretical point of view but most numerical experiences are confined to the periodic framework. Also the periodicity assumption allowed the extention of the homogenization theory to more general frameworks like non linear elasticity, p-laplacian, homogenization of general energy functionals. A problem of major importance in homogenization is the identification of bounds on the homogenized coefficients, known in the literature as the G-closure problem. It consists in describing the set of all possible tensors obtained by mixing a given number of base materials, in given proportions. Once identified such optimal bounds on homogenized 1