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

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.