Partizioni Ottime per il Problema di Yamabe

The aim of this thesis is to study the existence and regularity of optimal partitions for the Yamabe equation with an arbitrary number l ≥ 2 of components on a closed Riemannian manifold (M, g). Following a recent work of M. Clapp, A. Pistoia and H. Tavares, we consider a coupled competitive elliptic system of l equations and we provide sufficient geometric conditions that guarantee the existence of a least energy solution of the system. As the parameter that controls the interaction goes to infinity, the densities segregate, providing an optimal partition. Furthermore, we prove the existence of optimal G-invariant partitions for the Yamabe equation, where G is a compact group of isometries acting on M. In this setting, we also explore the case of three-term competitive interaction and we obtain analogous results. Moreover, we investigate the spectral properties of the three-term interaction and we provide a new Alt-Caffarelli-Friedman monotonicity formula involving three densities. Finally, we present some regularity results concerning the limiting profiles by N. Soave, H. Tavares, S. Terracini and A. Zilio. We establish the convergence in the Hölder space, Lipschitz continuity of the limit functions and we give a characterization of their common nodal set.