The optimization of the principal eigenvalue of indefinite weighted problems settled in bounded domains arises as a natural task in the study of the survival threshold for a species in population dynamics. We study the minimization of such eigenvalue, associated with Neumann boundary conditions, performing the analysis of the singular limit in case of arbitrarily small favourable region. We show that, in this regime, the favourable region is connected and it concentrates on the boundary of the domain. Though widely expected, these properties are still unknown in the general case. This is a joint research with Dario Mazzoleni and Gianmaria Verzini.