Weakly neutral inclusions of general shape II: Existence for small perturbations of balls
Hyeonbae Kang, Xiaofei Li, Shigeru Sakaguchi · arXiv (Cornell University) · 2019
An inclusion is said to be weakly neutral to uniform fields if it perturbs the uniform fields mildly upon insertion into the medium with uniform field, in comparison to the neutral inclusion which does not perturb the uniform field at all. The inclusion made of a pair of concentric balls can be made to be neutral, and hence weakly neutral, to uniform fields by choosing conductivity parameters appropriately, and it is the only known example of weakly neutral inclusions in three dimensions. We address the question regarding existence of weakly neutral inclusions of general shape in the form of the core-shell structure in three dimensions. We show, by the implicit function theorem on Banach spaces, that if the core is a small perturbation of a ball, then it can be coated or enclosed by a shell so that the resulting inclusion of the core-shell structure becomes weakly neutral to multiple uniform fields. The outer boundary of the shell is also a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work \cite{KLS2D} for two dimensions.