Singular Analysis of the Optimizers of the Principal Eigenvalue in Indefinite Weighted Neumann Problems

We study the minimization of the positive principal eigenvalue associated to aweighted Neumann problem settled in a bounded smooth domain \Omega \subset \BbbR N,N\geq 2, within a suitableclass of sign-changing weights. This problem arises in the study of the persistence of a species in pop-ulation dynamics. Denoting withuthe optimal eigenfunction and withDits superlevel set associatedto the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as themeasure ofDtends to zero. We show that, when the measure ofDis sufficiently small,uhas a uniquelocal maximum point lying on the boundary of \Omega andDis connected. Furthermore, the boundaryofDintersects the boundary of the box \Omega , and more precisely,\scrH N - 1(\partial D\cap \partial \Omega )\geq C| D| (N - 1)/Nforsome universal constantC >0. Though widely expected, these properties are still unknown if themeasure ofDis arbitrary