The Yamabe equation on a Riemannian manifold $(M,g)$ is relevant to the question of finding a constant scalar curvature metric on $M$ that is conformally equivalent to the given one.

An optimal $\ell$-partition for the Yamabe equation is a cover of $M$ by $\ell$ pairwise disjoint open subsets such that the Yamabe equation with Dirichlet boundary condition has a least energy solution on each one of these sets, and the sum of the energies of these solutions is minimal.

We will present some recent results obtained in collaboration with Angela Pistoia (La Sapienza Università di Roma) and Hugo Tavares (Universidade de Lisboa) that establish the existence and qualitative properties of such partitions.

If time allows, we will also present some results on symmetric optimal partitions obtained in collaboration with Angela Pistoia, and with Alberto Saldaña (UNAM) and Andrzej Szulkin (Stockholm University).