Quantitative Estimates on Periodic Homogenization of Nonlinear Elliptic Operators

Abstract: In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates $O(\varepsilon^{1/2})$ for a $C^{1,1}$ domain, and $O(\varepsilon^\sigma)$ for a Lipschitz domain, in which $\sigma\in(0,1/2)$ is close to zero. Based upon the convergence rates, an interior Lipschitz estimate, as well as a boundary H\"older estimate can be developed at large scales without any smoothness assumption, and these will implies reverse H\"older estimates established for a $C^1$ domain. By a real method developed by Z.Shen \cite{S3}, we consequently derive a global $W^{1,p}$ estimate for $2\leq p<\infty$. This work may be regarded as an extension of \cite{MAFHL,S5} to a nonlinear operator, and our results may be extended to the related Neumann boundary problems without any real difficulty.