  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.
