On s-harmonic functions on cones

Abstract

We deal with non negative functions satisfying \[ \left\{ \begin{array}{ll} (-Δ)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{array}\right. \] 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.

37 pages, 3 figures

References in corpus (1)

• On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition

Cited by in corpus (6)

• Some remarks about the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group
• Regularity of shape optimizers for some spectral fractional problems
• An anisotropic monotoncity formula, with applications to some segregation problems
• Segregated configurations involving the square root of the laplacian and their free boundaries
• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory
• Unique continuation from conical boundary points for fractional equations