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

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.