Homogenization of an obstacle problem with highly oscillating coefficients and obstacles

TL;DR

This paper introduces a viscosity-based homogenization approach for obstacle problems with rapidly oscillating obstacles and coefficients, capturing complex local behaviors and establishing a unique critical value.

Contribution

It develops a novel viscosity method for homogenizing obstacle problems with highly oscillating obstacles and coefficients, including the construction of a specialized corrector.

Findings

Constructed a corrector capturing singular behaviors near oscillating obstacles.

Proved the uniqueness of a critical value linking oscillations in coefficients and obstacles.

Established a homogenized operator for the obstacle problem with oscillations.

Abstract

We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.