# The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

## TL;DR

This paper investigates the detailed structure and regularity of the singular set in the thin obstacle problem for degenerate parabolic equations with a specific weight, extending classical monotonicity formulas to this context.

## Contribution

It provides a comprehensive analysis of the singular set in the thin obstacle problem for weighted degenerate parabolic equations, including new monotonicity formulas.

## Key findings

- Complete structure and regularity of the singular set established.
- Generalization of Almgren-Poon, Weiss, and Monneau monotonicity formulas.
- Application to the obstacle problem for the fractional heat operator.

## Abstract

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem 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 main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$).