Polarization tensor vanishing structure of general shape: Existence for small perturbations of balls

TL;DR

This paper demonstrates that small perturbations of a spherical domain can be constructed so that the resulting inclusion has a vanishing polarization tensor, extending known cases beyond perfect spheres.

Contribution

It proves the existence of non-spherical domains with vanishing polarization tensor through small perturbations of a sphere using the implicit function theorem.

Findings

Small perturbations of spheres can produce domains with vanishing polarization tensor.

The boundary of such domains can be described using spherical harmonics of degree zero and two.

This extends previous results from two dimensions to general shapes.

Abstract

The polarization tensor is a geometric quantity associated with a domain. It is a signature of the small inclusion's existence inside a domain and used in the small volume expansion method to reconstruct small inclusions by boundary measurements. In this paper, we consider the question of the polarization tensor vanishing structure of general shape. The only known examples of the polarization tensor vanishing structure are concentric disks and balls. We prove, by the implicit function theorem on Banach spaces, that a small perturbation of a ball can be enclosed by a domain so that the resulting inclusion of the core-shell structure becomes polarization tensor vanishing. The boundary of the enclosing domain is given by a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work \cite{KLS2D} for two dimensions.