Quantitative homogenization for the obstacle problem and its free boundary

Abstract

In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.

Quantitative convergence of the “bulk” free boundary in an oscillatory obstacle problem

“…The works that are closest to ours are [12] and [1]. Our results quantify the qualitative convergence results obtained in [12], and we go further by establishing the convergence of the bulk free boundary of the oscillatory obstacle problem to the free boundary of the unperturbed problem.…”

“…Our results quantify the qualitative convergence results obtained in [12], and we go further by establishing the convergence of the bulk free boundary of the oscillatory obstacle problem to the free boundary of the unperturbed problem. The recent work [1] establishes a large scale regularity theory for the obstacle problem with an oscillatory divergence-form elliptic operator. They make an assumption on "compatibility" of the obstacle with the operator, which avoids the kind of oscillatory contact set that we study here.…”