Uniform estimate for strongly elliptic equations in high-contrast composites

This paper considers uniform estimates for strongly elliptic equations in high-contrast composites. The composites consist of an $ {\epsilon} $-periodic lattice of fibers with high conductivity, included in a connected material with normal conductivity. The diffusion coefficients of the elliptic equations, depending on the conductivities, are not bounded above. The equations have fast diffusion inside the fibers and slow diffusion elsewhere. Let $ \omega^2\in(1, \infty) $ denote the conductivity ratio of the fibers to the connected material and let $ {\epsilon}\frac \mu 2\in(0, \frac 12) $ be the diameter of the horizontal cross-section of each fiber. This work presents $ W^{1, p} $ estimates uniformly in $ {\epsilon}, \omega, \mu $ for the solutions of the elliptic equations.