Yamabe systems and optimal partitions on manifolds with symmetries

TL;DR

This paper establishes the existence of symmetric optimal partitions for the Yamabe equation on manifolds with group symmetries by analyzing a related elliptic system and its limiting behavior.

Contribution

It introduces a novel approach using a coupled elliptic system to find symmetric optimal partitions for the Yamabe problem on manifolds with symmetries.

Findings

Existence of regular G-invariant partitions with multiple components.

Construction of least energy G-invariant solutions with nontrivial components.

Limit profiles of solutions form optimal partitions as competition parameter tends to -infinity.

Abstract

We prove the existence of regular optimal $G$-invariant partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$ when $G$ is a compact group of isometries of $M$ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy $G$-invariant solution with nontrivial components and we show that the limit profiles of the its components separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing $G$-invariant solution to the Yamabe equation with precisely two nodal domains.