Optimal regularity for the no-sign obstacle problem

TL;DR

This paper establishes the optimal $C^{1,1}$ regularity for a broad class of obstacle problems with minimal assumptions on the data, and proves the local smoothness of the free boundary under certain conditions.

Contribution

It proves the optimal regularity for obstacle problems with minimal assumptions and shows free boundary regularity under thickness and Dini conditions.

Findings

Proved $C^{1,1}$ regularity under the weakest possible assumptions.

Established local $C^1$-smoothness of the free boundary.

Solved a long-standing open problem in obstacle problem regularity.

Abstract

In this paper we prove the optimal $C^{1,1}(B_\frac12)$-regularity for a general obstacle type problem $$ \lap u = f\chi_{\{u\neq 0\}}\textup{in $B_1$}, $$ under the assumption that $f*N$ is $C^{1,1}(B_1)$, where $N$ is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$-regularity. As a by-product of the $C^{1,1}$-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$, the free boundary is locally a $C^1$-graph close to $x^0$, provided $f$ is Dini. This completely settles the question of the optimal regularity of this problem, that has been under much attention during the last two decades.