Phase separation, optimal partitions, and nodal solutions to the Yamabe equation on the sphere

TL;DR

This paper investigates the relationship between optimal partitions and sign-changing solutions to the Yamabe equation on the sphere, establishing existence results through analysis of elliptic systems and symmetry considerations.

Contribution

It introduces a novel correspondence between optimal partitions and least-energy sign-changing solutions with specified nodal domains for the Yamabe equation on the sphere.

Findings

Existence of optimal partitions with symmetry constraints.

Correspondence between solutions and partitions with M nodal domains.

Analysis of limit profiles of elliptic systems.

Abstract

We study an optimal M-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least-energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely M nodal domains. The existence of an optimal partition is established through the study of the limit profiles of least-energy solutions to a weakly coupled competitive elliptic system on the sphere.