Polarization tensors via homogenization theory

Polarization tensors corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction. A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is continuous as the volume fraction tends to zero. Our approach has its own advantages. In particular, it provides another method to deduce optimal estimates on polarization tensors in any dimension from those on homogenized tensors, along with the information on underlying microstructures.

People: Tuhin Ghosh, Venkateswaran P. Krishnan and M. Vanninathan
Status: Completed, Paper to be submitted