Quantitative convergence of the "bulk'' free boundary in an oscillatory obstacle problem

TL;DR

This paper analyzes the convergence behavior of the free boundary in an oscillatory obstacle problem, providing a quantitative rate of convergence to the classical free boundary under regularity assumptions.

Contribution

It establishes a linear convergence rate for a regularized free boundary to the classical free boundary in an oscillatory obstacle problem.

Findings

Convergence rate is linear in the minimal length scale.

Provides a quantitative estimate for free boundary convergence.

Assumes regularity of the classical obstacle problem's free boundary.

Abstract

We consider an oscillatory obstacle problem where the coincidence set and free boundary are also highly oscillatory. We establish a rate of convergence for a regularized notion of free boundary to the free boundary of a corresponding classical obstacle problem, assuming the latter is regular. The convergence rate is linear in the minimal length scale determined by the fine properties of a corrector function.