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

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.' " .x/ WD ' 0 .x/ C " p .x="/: