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On $s$-harmonic
functions on cones
Susanna Terracini, Giorgio Tortone and Stefano Vita |
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Vol. 11 (2018), No. 7, 1653–1691
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Abstract |
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We deal with nonnegative functions satisfying where and is a given cone on with vertex at zero. We consider the case when approaches , 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. |
Keywords
fractional Laplacian, conic functions, asymptotic behavior,
Martin kernel
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Mathematical Subject Classification 2010
Primary: 35R11
Secondary: 35B45, 35B08
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Milestones
Received: 18 May 2017
Revised: 18 September 2017
Accepted: 18 February 2018
Published: 20 May 2018
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Authors |
| Susanna Terracini | |
| Dipartimento di Matematica “Giuseppe
Peano” Università di Torino Torino Italy |
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| Giorgio Tortone | |
| Dipartimento di Matematica “Giuseppe
Peano” Università di Torino Torino Italy |
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| Stefano Vita | |
| Dipartimento di Matematica “Giuseppe
Peano” Università di Torino Torino Italy |
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