Fractional Q-curvature on the sphere and optimal partitions

2025/04/23 by Héctor A. Chang‐Lara, Chang-Lara, Héctor A., Juan Carlos Fernández +3 · 1 citation
Mathematics · Physics and Astronomy · #35B33 #35R01 #35R11 #35S15 #58J40 #58J70 #58J90 #Advanced Banach Space Theory #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Fixed Point Theorems Analysis

paper · pdf · doi:10.48550/arxiv.2504.16882

openalex publication_date 2025/04/23 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional Q-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

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• The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces

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