We deal with non negative functions which are s-harmonic on a given cone of the n-dimensional Euclidean space with vertex at zero, vanishing on the complementary. We consider the case when the parameter 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.

Vita, S. (2019). On s-harmonic functions on cones funzioni s-armoniche su coni. Intervento presentato a: Bruno Pini Mathematical Analysis Seminar, Bologna, Italia [10.6092/issn.2240-2829/10366].

On s-harmonic functions on cones funzioni s-armoniche su coni

Vita, S
2019

Abstract

We deal with non negative functions which are s-harmonic on a given cone of the n-dimensional Euclidean space with vertex at zero, vanishing on the complementary. We consider the case when the parameter 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.
paper
Asymptotic behavior; Conic functions; Fractional Laplacian; Martin kernel;
English
Bruno Pini Mathematical Analysis Seminar
2018
2019
10
1
28
41
reserved
Vita, S. (2019). On s-harmonic functions on cones funzioni s-armoniche su coni. Intervento presentato a: Bruno Pini Mathematical Analysis Seminar, Bologna, Italia [10.6092/issn.2240-2829/10366].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/282752
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