Reverse Holder's inequality in non linear conductivity

Abstract

We establish lower bounds for the overall conductivity of a class of non linear composites. The composites are made of an arbitrary number of anisotropic phases. The local 'density of energy' is subquadratic. This problem cannot be treated by most methods considered in the existing literature such as the well known generalization of the linear Hashin-Shtrikman method due to Willis [33] and developed by Talbot & Willis [30]. Very recently, Talbot and Willis have developed a new method based on certain properties of BMO functions [31], [32]. Their calculations apply when the phases are isotropic. However when at least one of the phases is not isotropic, the only result available, prior to the present work, was the classical Wiener bound. We develop yet another method which is completely different from that of Talbot and Willis. It is based on the idea of using an appropriate reverse Holder inequality. The main mathematical tools come from the theory of planar quasiconformal mappings. We use results due to Astala [1] and Eremenko and Hamilton [11]. Our new bounds apply, under certain hypotheses, to two dimensional problems. When they apply they are always at least as good as those of Wiener. We exhibit examples in which our bounds are strictly better.