Strongly elliptic equations with periodic coefficients in two-dimensional space

Regularity for strongly elliptic equations with highly oscillatory \varepsilon -periodic coefficients in two-dimensional space is addressed. In each \varepsilon -cell, the diffusion coefficients of the elliptic equations are \omega^{2}\in(1,\infty) in a small disk with radius \frac{\varepsilon\mu}{4}\,(<\frac{1}{4}) and 1 outside the disk of the cell. Two cases are considered. Case one is that \varepsilon , \mu , \omega are independent in the elliptic equations. So, the diffusion coefficients of the elliptic equations are \varepsilon -periodic and discontinuous. L^{p} -gradient estimate uniformly in \varepsilon , \mu , \omega for the elliptic solutions is derived. However, the integrability p\,(>2) of the solutions is not a large number. Case two is that \varepsilon , \mu\, (=\omega^{-1}) are independent in the elliptic equations. The diffusion coefficients of the elliptic equations are \varepsilon -periodic, discontinuous, and L^{1} -bounded. Lipschitz estimate uniformly in \varepsilon , \mu\,(=\omega^{-1}) for the elliptic solutions is obtained.