On the fine regularity of the singular set in the nonlinear obstacle problem

TL;DR

This paper refines the understanding of the singular set's regularity in the nonlinear obstacle problem, showing a detailed stratification into well-behaved and lower-dimensional parts, extending classical results.

Contribution

It improves previous regularity results by decomposing the singular set into manifolds and lower-dimensional anomalies, enhancing the geometric understanding of free boundaries.

Findings

Singular set decomposed into 'good' and 'anomalous' parts.

'Good' part covered by $C^{1,1-}$ manifolds.

'Anomalous' part has lower dimension.

Abstract

We revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem, we show that each stratum can be further decomposed into a `good' part and an `anomalous' part, where the former is covered by $C^{1,1-}$ manifolds, and the latter is of lower dimension.