Non-local homogenized limits for composite media with highly anisotropic periodic fibres

We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.