# The regular free boundary in the thin obstacle problem for degenerate parabolic equations

## TL;DR

This paper investigates the existence, optimal regularity, and free boundary regularity near regular points in a thin obstacle problem related to the fractional heat operator, with a focus on local estimates and blowup analysis.

## Contribution

It establishes local regularity results and the boundedness of the time-derivative, enabling reduction to elliptic problems and advancing understanding of the free boundary in fractional parabolic equations.

## Key findings

- Proved existence and regularity of solutions near regular points.
- Established $H^{1+eta, (1+eta)/2}$ regularity of the free boundary.
- Demonstrated the boundedness of the solution's time-derivative.

## Abstract

In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $H^{1+\gamma,\frac{1+\gamma}{2}}$ regularity of the free boundary near such regular points.