Inverse Diffusivity Problem via Homogenization Theory

TL;DR

This paper explores the relationship between polarization tensors and homogenized tensors for inhomogeneities with positive volume fractions, providing new insights and methods for inverse problems in homogenization theory.

Contribution

It introduces a polarization tensor linked to positive volume inhomogeneities and relates it to the homogenized tensor, offering a novel approach to inverse problems.

Findings

Established a relation between polarization and homogenized tensors.

Analyzed the continuity of the polarization tensor as volume fraction approaches zero.

Provided methods to derive optimal estimates on polarization tensors from homogenized tensors.

Abstract

Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. 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, as we will see. 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.