Existence of weakly neutral coated inclusions of general shape in two dimensions

A two-dimensional inclusion of core–shell structure is neutral to multiple uniform fields if and only if the core and the shell are concentric disks, provided that the conductivity of the matrix is isotropic. An inclusion is said to be neutral if upon its insertion the uniform field is not perturbed at all. In this paper, we consider inclusions of core–shell structure of general shape which are weakly neutral to multiple uniform fields. An inclusion is said to be weakly neutral if the field perturbation is mild. We show, by an implicit function theorem, that if the core is a small perturbation of a disk, then we can coat it by a shell so that the resulting structure becomes weakly neutral to multiple uniform fields.