The structure of the regular part of the free boundary close to singularities in the obstacle problem

TL;DR

This paper investigates the detailed asymptotic behavior of the regular free boundary near singularities in the obstacle problem, advancing understanding of the boundary's structure and contributing to longstanding conjectures.

Contribution

It provides the first detailed asymptotic analysis of the regular free boundary near singularities and offers partial classifications of global solutions with unbounded coincidence sets.

Findings

First asymptotic description of the free boundary near singularities

Partial classification of global solutions with unbounded sets

Progress towards longstanding conjectures in obstacle problems

Abstract

We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets in higher dimensions.