Homogenization of the Eigenvalues of the Neumann–Poincaré Operator

Abstract

In this article, we investigate the spectrum of the Neumann–Poincare operator $${{\mathcal {K}}}_\varepsilon ^*$$ (or equivalently, that of the associated Poincare variational operator $$T_\varepsilon $$ ) associated to a periodic distribution of small inclusions with size $$\varepsilon $$ , and its asymptotic behavior as the parameter $$\varepsilon $$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to $$\varepsilon $$ . This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light on the issue of the homogenization of the voltage potential $$u_\varepsilon $$ caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity a, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of $$u_\varepsilon $$ is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of $$u_\varepsilon $$ is always possible when a is either positive, or negative with a ‘small’ or ‘large’ modulus.