Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients

In this work, we are interested in the homogenization of time-harmonic Maxwell’s equations a composite medium with periodically distributed small inclusions negative material. Here material is modelled by permittivity and permeability. Due to sign-changing coefficients equations, it not straightforward obtain uniform energy estimates apply usual techniques. The goal article explain how proceed context. analysis based on precise study two associated scalar problems: one involving Dirichlet boundary conditions, another permeability Neumann conditions. For both problems, criterion physical parameters ensuring invertibility corresponding operators as size tends zero. process, link existing so-called Neumann–Poincaré operator complementing literature topic. Then use results obtained for problems derive system. At stage, an additional difficulty comes from fact that also sign-indefinite due term frequency. To cope it, establish some sort compactness result.