We will discuss some recent results concerning weighted eigenvalue problems in bounded Lipschitz domains, under Neumann boundary conditions.

The optimization of the distribution of resources leads to minimize a principal eigenvalue with respect to the sign-changing weight. Important qualitative properties of the positivity set of the optimal weight, such as being connected, as well as its location, are still not known in general.

We will present some new achievements in the asymptotical study regarding these properties.

Joint works with Dario Mazzoleni (Università di Pavia), Lorenzo Ferreri (Scuola Normale Superiore di Pisa) and Gianmaria Verzini (Politecnico di Milano).