Homogenization of oblique boundary value problems

TL;DR

This paper develops quantitative homogenization results for nonlinear Neumann problems with oscillating coefficients in half-spaces, including cases with general normal directions, and establishes Hölder continuity of boundary data in rotation-invariant cases.

Contribution

It extends homogenization theory to nonlinear Neumann problems with oscillations in general directions and proves boundary regularity results for the homogenized operator.

Findings

Quantitative homogenization results for oscillating nonlinear Neumann problems.

Hölder continuity of the homogenized boundary data in rotation-invariant cases.

Applicability to problems in half-spaces with arbitrary normal directions.

Abstract

We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the H\"{o}lder continuity of the homogenized boundary data.