Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

TL;DR

This paper studies the homogenization of fully nonlinear PDEs with oscillatory Neumann boundary data in general domains, showing uniform convergence to a homogenized solution and continuity of the homogenized boundary data.

Contribution

It extends homogenization results to nonlinear PDEs with oscillatory boundary conditions in non-flat domains, under rotation invariance assumptions.

Findings

Solutions converge uniformly to the homogenized solution.

Homogenized Neumann data is continuous with respect to boundary normal.

Results apply to general domains without flat boundary parts.

Abstract

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary.