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arXiv · 2504.16882

Fractional $Q$-curvature on the sphere and optimal partitions

Héctor A. Chang-Lara·Juan Carlos Fernández·Alberto Saldaña
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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 H\"older 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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Héctor A. Chang-Lara, Juan Carlos Fernández, Alberto Saldaña. 2025-04-23. Fractional $Q$-curvature on the sphere and optimal partitions. https://arxiv.org/abs/2504.16882

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