  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.
