Generic regularity of free boundaries for the thin obstacle problem

TL;DR

This paper proves that for generic solutions to the thin obstacle problem, the degenerate set of the free boundary is very small in measure, leading to generic smoothness of the entire free boundary in low dimensions.

Contribution

It establishes that the degenerate set has zero measure in a specific sense for generic solutions, confirming smoothness of free boundaries in low dimensions and solving a conjecture in the field.

Findings

Degenerate set has zero ^{n-3-} measure for generic solutions.

For dimensions  , the free boundary is generically smooth.

Confirms a conjecture of Schaeffer in ^3 and ^4.

Abstract

The free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most $(n-2)$-dimensional. Our main result establishes that, in fact, the degenerate set has zero $\mathcal{H}^{n-3-\alpha_0}$ measure for a generic solution. As a by-product, we obtain that, for $n+1 \leq 4$, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in $\mathbb{R}^3$ and $\mathbb{R}^4$ for the thin obstacle problem.