Sharp regularity for singular obstacle problems

TL;DR

This paper establishes sharp local regularity results for solutions to singular obstacle problems, including optimal free boundary regularity, extending known results even in the linear case.

Contribution

It provides the first sharp regularity results for singular obstacle problems with explicit free boundary regularity in terms of problem parameters.

Findings

Sharp local $C^{1,eta}$ regularity for solutions

Optimal $C^{1, au}$ regularity at the free boundary

Regularity results valid even in the linear case

Abstract

We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for $0<\gamma<1$ and $p\ge2$. At the free boundary $\partial\{u>\varphi\}$, we prove optimal $C^{1,\tau}$ regularity of solutions, with $\tau$ given explicitly in terms of $p$, $\gamma$ and smoothness of $\varphi$, which is new even in the linear setting.