Regularity for strongly elliptic equations with ϵ-periodic highly oscillatory coefficients in two-dimensional space is concerned. In each ϵ-cell, the diffusion coefficients of the elliptic equations are ω 2 ∈ ( 1 , ∞ ) in a small disk with radius ϵµ 4 ( < 1 4 ) and 1 outside the disk of the cell. Two cases are considered. Case one is that ϵ,µ,ω are independent in the elliptic equations. So the diffusion coefficients of the elliptic equations are ϵ-periodic and discontinuous. L p -gradient estimate uniform in ϵ,µ,ω for the elliptic solutions is derived. However, the integrability p ( >2) of the solutions is not a large number. Case two is that ϵ , µ ( = ω − 1 ) are independent in the elliptic equations. The diffusion coefficients of the elliptic equations are ϵ-periodic, discontinuous, and L 1 -bounded. Lipschitz estimate uniform in ϵ , µ ( = ω − 1 ) for the elliptic solutions is obtained.