An over-determined boundary value problem arising from neutrally coated inclusions in three dimensions

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We consider the neutral inclusion problem in three dimensions which is to prove if a coated structure consisting of a core and a shell is neutral to all uniform fields, then the core and the shell must be concentric balls if the matrix is isotropic and confocal ellipsoids if the matrix is anisotropic. We first derive an over-determined boundary value problem in the shell of the neutral inclusion, and then prove in the isotropic case that if the over- determined problem admits a solution, then the core and the shell must be concentric balls. As a consequence it is proved that the structure is neutral to all uniform fields if and only if it consists of concentric balls provided that the coefficient of the core is larger than that of the shell.