Uniform integrability in periodic homogenization of fully nonlinear elliptic equations

Abstract

This paper is devoted to the study of uniform \(W^{1,\frac{np}{n-p}}\)- and \(W^{2,p}\)-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.