Calculus of Variations and Geometric Measure Theory

S. Terracini - G. Tortone - S. Vita

On s-harmonic functions on cones

created by tortone on 01 Nov 2023


Published Paper

Inserted: 1 nov 2023
Last Updated: 1 nov 2023

Journal: Analysis & PDE
Year: 2018
Doi: 10.2140/apde.2018.11.1653

ArXiv: 1705.03717 PDF


We deal with non negative functions satisfying \[ \left\{ \begin{array}{ll} (-\Delta)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{array}\right. \] where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half sphere. These conic 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.