# $C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem

## TL;DR

This paper proves that the free boundary in a one-phase fractional Laplacian problem is infinitely smooth, extending previous regularity results from $C^{1, ext{alpha}}$ to $C^{ ext{infinity}}$ for all $0<s<1$.

## Contribution

The authors establish $C^{ ext{infinity}}$ regularity of the free boundary for the fractional Laplacian problem, generalizing prior $C^{1, ext{alpha}}$ results and extending known results for the case $s=1/2$.

## Key findings

- Free boundary is $C^{ ext{infinity}}$ smooth.
- Extension of regularity results from $s=1/2$ to all $0<s<1$.
- Improved understanding of free boundary regularity in fractional problems.

## Abstract

We consider a one-phase free boundary problem involving fractional Laplacian $(-\Delta)^s$, $0<s<1$. D. De Silva, O. Savin, and Y. Sire proved that the flat boundaries are $C^{1,\alpha}$. We raise the regularity to $C^{\infty}$, extending the result known for $(-\Delta)^{1/2}$ by D. De Silva and O. Savin.