We deal with nonnegative functions satisfying where s ∈(0, 1) and C is a given cone on ℝn 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
Terracini, S., Tortone, G., Vita, S. (2018). On s-Harmonic functions on cones. ANALYSIS & PDE, 11(7), 1653-1691 [10.2140/apde.2018.11.1653].
On s-Harmonic functions on cones
Terracini, S;Vita, S
2018
Abstract
We deal with nonnegative functions satisfying where s ∈(0, 1) and C is a given cone on ℝn 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 diffusionsI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.