Uniform Integrability in Periodic Homogenization of Fully Nonlinear Equations

TL;DR

This paper establishes uniform integrability estimates for viscosity solutions in periodic homogenization of fully nonlinear elliptic equations, advancing understanding without regularity assumptions and applicable to non-convex functionals.

Contribution

It provides new uniform integrability estimates for solutions to fully nonlinear homogenization problems without regularity assumptions, covering non-convex functionals.

Findings

Uniform $W^{1,rac{np}{n-p}}$- and $W^{2,p}$-estimates established

Characterization of the effective Hessian and gradient sizes

Estimates are valid even for non-convex functionals

Abstract

This paper is devoted to the study of uniform $W^{1,\frac{np}{n-p}}$- and $W^{2,p}$-estimates for viscosity solutions to fully nonlinear, uniformly elliptic, periodic homogenization problems, up to boundaries, subject to Dirichlet boundary conditions. We characterize the size of "effective" Hessian and gradient of viscosity solutions to homogenization problems, and prove its uniform integrability without any regularity assumption on the governing functionals. Our estimates are new even for the standard problems. Our analysis applies to a large class of non-convex functionals.