Neumann Homogenization via Integro-Differential Operators, Part 2: singular gradient dependence

TL;DR

This paper advances the homogenization theory for Neumann problems with singular gradient dependence by employing integro-differential methods, extending previous results to more complex boundary conditions and nonlinear cases.

Contribution

It introduces a novel integro-differential approach to Neumann homogenization with singular drift, including non-co-normal oscillatory conditions, and connects to existing nonlinear results.

Findings

Developed an integro-differential homogenization framework for boundary problems.

Extended homogenization results to singular gradient-dependent Neumann conditions.

Recast existing fully nonlinear Neumann homogenization results within this new framework.

Abstract

We continue the program initiated in a previous work, of applying integro-differential methods to Neumann Homogenization problems. We target the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with \emph{non-co-normal} oscillatory Neumann conditions. Our analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. Also, we use homogenization results for regular Dirichlet problems to build barriers for the oscillatory Neumann problem with the singular gradient term. We note that our method allows to recast some existing results for fully nonlinear Neumann homogenization into this same framework. This version is the journal version.