On s-harmonic functions on cones

where $s \in (0, 1)$ and $C$ is a given cone on $ℝ^{n}$ 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.

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Keywords

fractional Laplacian, conic functions, asymptotic behavior, Martin kernel

Mathematical Subject Classification 2010

Primary: 35R11

Secondary: 35B45, 35B08

Milestones

Received: 18 May 2017

Revised: 18 September 2017

Accepted: 18 February 2018

Published: 20 May 2018

Authors

Susanna Terracini
Dipartimento di Matematica “Giuseppe Peano” Università di Torino Torino Italy

Giorgio Tortone
Dipartimento di Matematica “Giuseppe Peano” Università di Torino Torino Italy

Stefano Vita
Dipartimento di Matematica “Giuseppe Peano” Università di Torino Torino Italy

Vol. 11, No. 7, 2018

Susanna Terracini, Giorgio Tortone and Stefano Vita

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors