Optimal pinwheel partitions for the Yamabe equation

TL;DR

This paper proves the existence of optimal symmetric partitions for the Yamabe equation in the entire space, using solutions to a coupled system that segregate as the coupling parameter tends to minus infinity, and finds infinitely many sign-changing solutions.

Contribution

It introduces a novel approach linking coupled Yamabe systems with optimal partitions and establishes the existence of multiple sign-changing solutions with symmetry properties.

Findings

Existence of an optimal partition made of mutually isometric, symmetry-invariant sets.

Solutions to a coupled Yamabe system segregate as coupling tends to minus infinity.

Existence of infinitely many new sign-changing solutions.

Abstract

We establish the existence of an optimal partition for the Yamabe equation in the whole space made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to minus infinity, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation that are different from those previously found in the by W.Y. Ding, and del Pino, Musso, Pacard and Pistoia