Reverse Holder's inequality in non linear conductivity |
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Authors: | V Nesi |
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Institution: | (1) Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW, UK;(2) Dipartimento di Matematica Pura & Applicata, 67010 L'Aquila, Italy, IT |
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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.
Received October 29, 1995 |
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