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A two dimensional inclusion of core-shell structure is neutral to multipleuniform 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 saidto 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 shapewhich are weakly neutral to multiple uniform fields. An inclusion is said to beweakly neutral if the field perturbation is mild. We show, by an implicitfunction theorem, that if the core is a small perturbation of a disk then wecan coat it by a shell so that the resulting structure becomes weakly neutralto multiple uniform fields.

Existence of coated inclusions of general shape weakly neutral to multiple fields in two dimensions