Characterizing compact coincidence sets in the obstacle problem -- a short proof

TL;DR

This paper provides a concise proof that bounded coincidence sets with nonempty interior in the classical obstacle problem are ellipsoids, advancing understanding of the structure of solutions in higher dimensions.

Contribution

It offers a simplified, extendable proof of the characterization of bounded coincidence sets as ellipsoids in the obstacle problem.

Findings

Bounded coincidence sets with interior are ellipsoids.

The proof is concise and easily extendable.

Supports ongoing research on unbounded coincidence sets.

Abstract

Motivated by the almost completely open problem of characterizing unbounded coincidence sets of global solutions of the classical obstacle problem in higher dimensions, we give in this note a concise and easy-to-extend proof of the known fact that if the coincidence set $\{u=0 \}$ of a global solution $u$ is bounded with nonempty interior then it is an ellipsoid.