Abstract
This paper addresses a question regarding optimal bounds of the polarization tensors. We prove that the geometry independent optimal bounds for the trace of the polarization tensor can be improved if the domain has a certain thickness. This paper addresses a question regarding optimal bounds of the polarization tensors (PT). We refer readers to a review article (2) in these proceedings for the definition and properties of the PT and its recent applications to inverse problems and eective medium theory. Based on variational techniques first introduced by Hashin and Shtrikman (5), and further described in (6), Lipton (7) and Capdeboscq-Vogelius (3) obtained ge- ometry independent optimal bounds of Hashin-Shtrikman (HS) type for the trace of the PT. Let M denote the PT associated with the bounded domain D R d whose volume |D| = 1. We suppose that D has the constant conductivity 0 < 1 < +1 while the background medium R d D has 0 < 1. Then, for d = 2,3, HS bounds are as follows 1 :
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