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Description

We deal with functions satisfying \begin{equation}\label{P_C} \begin{cases} (-\Delta)^s u_s=0 & \mathrm{in}\quad C, \\ u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{cases} \end{equation} where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We are mainly concerned with the case when $s$ approaches $1$. These functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions. This is a joint work with Giorgio Tortone and Stefano Vita.