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2018 On $s$-harmonic functions on cones
Susanna Terracini, Giorgio Tortone, Stefano Vita
Anal. PDE 11(7): 1653-1691 (2018). DOI: 10.2140/apde.2018.11.1653

Abstract

We deal with nonnegative functions satisfying

( Δ ) s u s = 0  in  C , u s = 0  in  n C ,

where s ( 0 , 1 ) and C is a given cone on 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.

Citation

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Susanna Terracini. Giorgio Tortone. Stefano Vita. "On $s$-harmonic functions on cones." Anal. PDE 11 (7) 1653 - 1691, 2018. https://doi.org/10.2140/apde.2018.11.1653

Information

Received: 18 May 2017; Revised: 18 September 2017; Accepted: 18 February 2018; Published: 2018
First available in Project Euclid: 15 January 2019

zbMATH: 1391.35401
MathSciNet: MR3810469
Digital Object Identifier: 10.2140/apde.2018.11.1653

Subjects:
Primary: 35R11
Secondary: 35B08, 35B45

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 7 • 2018
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