Regularity of the singular set in the fully nonlinear obstacle problem

TL;DR

This paper studies the structure of the singular set in a fully nonlinear obstacle problem, showing stratification and regularity properties that generalize known results for the Laplacian case.

Contribution

It establishes the stratification and regularity of the singular set for convex fully nonlinear elliptic operators, extending previous Laplacian-based results.

Findings

Top stratum is covered by a $C^{1,eta}$-manifold.

Lower strata are covered by $C^{1, ext{log}^eta}$-manifolds.

Generalizes regularity results from Laplacian to fully nonlinear operators.

Abstract

For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\varepsilon}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.