Strongly elliptic equations with periodic coefficients in two-dimensional space

Abstract

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