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 lambda = 0 by a newly derived variant of high-contrast Poincare-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.