Complete classification of global solutions to the obstacle problem

TL;DR

This paper conclusively classifies all global solutions to the obstacle problem in Euclidean space, showing their coincidence sets are limited to specific geometric shapes such as half-spaces, ellipsoids, paraboloids, or cylinders with these bases.

Contribution

It proves the long-standing conjecture that the coincidence set of global solutions must be one of a few specific geometric forms, completing the classification.

Findings

Coincidence sets are either half-spaces, ellipsoids, paraboloids, or cylinders with these bases.

The classification resolves a 90-year-old open problem in the theory of obstacle problems.

Provides a complete geometric characterization of global solutions.

Abstract

The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.