Bounding the current in nonlinear conducting composites

Suppose a three-dimensional composite of two nonlinear conducting phases mi xed in fixed proportions is subject to a fixed average electric field. What values can the average current take as the microstructure varies over all configurations? What microstructures produce the maximum or minimum current flow? Which microstructures are best for guiding the current in a given di rection? Here, following the compensated compactness method of Tartar (1977 : Estimation de coefficients homogeneises. In: Glowinski, R., Lions, J.-L. (Eds.), Computer Methods in Applied Sciences and Engineering, Springer-Verl ag Lecture Notes in Mathematics 704. Springer-Verlag, Berlin, pp. 136-212) we show how one can obtain remarkably tight bounds on the average current f low. In many, but not all cases, we find that simple laminate structures pr oduce the maximum or minimum current flow, and are best for guiding the cur rent in a given direction. Sometimes it is advantageous to orient the layer surfaces parallel (rather than orthogonal) to the direction of the applied field to generate the minimum current flow in that direction. (C) 2000 Els evier Science Ltd. All rights reserved.