Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities

In earlier work [8] we have derived a very general representation formula for the voltage perturbation caused by volumetrically small inhomogeneities in an otherwise known conductor. The ingredients of this formula are (1) a limiting probability measure, (2) a “background” fundamental solution, and (3) an “effective” polarization tensor. In [9] we proved optimal average bounds on the very essential polarization tensor, and we used these bounds to obtain sharp estimates for the volume of the inhomogeneities in terms of a few boundary voltage measurements. The principal goal of this paper is to prove that these optimal average bounds actually hold in a pointwise sense. A second goal is to show how the optimal pointwise bounds may be used to establish a particular, very specific form of the representation formula for thin inhomogeneities. This “thin inhomogeneity