Open AccessFractional perimeters on the sphere
Author(s) -
Andreas Kreuml,
Olaf Mordhorst
Publication year - 2021
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021083
Subject(s) - isoperimetric inequality , mathematics , perimeter , combinatorics , limit (mathematics) , geometry , mathematical analysis
This note treats several problems for the fractional perimeter or \begin{document}$ s $\end{document} -perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as \begin{document}$ s \nearrow 1 $\end{document} is proven. It is shown that their limit as \begin{document}$ s \searrow -\infty $\end{document} can be expressed in terms of the volume.