CAMGSD, IST, Universidade de Lisboa
ANAP group seminar
We give conditions for the existence of regular optimal partitions, with an arbitrary number l≥ 2 of components, for the Yamabe equation on a closed Riemannian manifold (M, g). To this aim, we study a weakly coupled competitive elliptic system of equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if dim M ≥ 10, (M, g) is not locally conformally flat and satisfies an additional geometric assumption whenever dim M = 0. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to infinity, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For l≥ 2, the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.
This talk is based on a joint work with M. Clapp (UNAM) and A. Pistoia (Roma La Sapienza).