Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions

TL;DR

This paper advances the understanding of periodic homogenization for fully nonlinear second-order PDEs in half-space domains with Neumann boundary conditions, focusing on singular problems requiring boundary condition determination.

Contribution

It provides new results for nonlinear equations and boundary conditions, extending prior linear and restrictive nonlinear homogenization work to more general settings.

Findings

Established homogenization results for fully nonlinear PDEs in half-space domains.

Solved ergodic problems related to boundary conditions in structured domains.

Extended previous linear and nonlinear homogenization theories.

Abstract

We study periodic homogenization problems for second-order pde in half-space type domains with Neumann boundary conditions. In particular, we are interested in "singular problems" for which it is necessary to determine both the homogenized equation and boundary conditions. We provide new results for fully nonlinear equations and boundary conditions. Our results extend previous work of Tanaka in the linear, periodic setting in half-spaces parallel to the axes of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic framework. The key step in our analysis is the study of associated ergodic problems in domains with similar structure.