# Improved regularity for the porous medium equation along zero level-sets

## TL;DR

This paper establishes sharp regularity estimates for solutions of the porous medium equation specifically along their zero level-sets, improving understanding of free boundary behavior under certain conditions.

## Contribution

It introduces new regularity results for the porous medium equation along zero level-sets, utilizing approximation and localization techniques from heat equation analysis.

## Key findings

- Solutions are locally of class C^{1-,1/2-} along free boundary points
- Regularity results hold under a proximity regime on the nonlinearity exponent
- Method involves importing information from the heat equation through approximation

## Abstract

In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. Then, we prove that solutions are locally of class $\mathcal{C}^{1-,\frac{1}{2}-}$ along free boundary points $x_0\in \partial\left\lbrace u>0\right\rbrace$, both in time and space. Our argument consists of importing information from the heat equation, through approximation and localization methods.