arXiv subjects
Publications and source records attributed to Runcao Lyu.
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.
In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are needed when classifying the half-space $(2k-\frac{1}{2})$-homogeneous solutions to the thin obstacle problems in $\mathbb{R}^3$. It is a generalization of one part of Savin-Yu's work \cite{savin2021halfspace} on classifying the half-space $\frac{7}{2}$-homogeneous solutions.