Singular Analysis of the Optimizers of the Principal Eigenvalue in Indefinite Weighted Neumann Problems
Dario Mazzoleni, Benedetta Pellacci, Gianmaria Verzini · SIAM Journal on Mathematical Analysis · 2023
Abstract. We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain [Formula: see text], [Formula: see text], within a suitable class of sign-changing weights. This problem arises in the study of the persistence of a species in population dynamics. Denoting with [Formula: see text] the optimal eigenfunction and with [Formula: see text] its superlevel set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of [Formula: see text] tends to zero. We show that, when the measure of [Formula: see text] is sufficiently small, [Formula: see text] has a unique local maximum point lying on the boundary of [Formula: see text] and [Formula: see text] is connected. Furthermore, the boundary of [Formula: see text] intersects the boundary of the box [Formula: see text], and more precisely, [Formula: see text] for some universal constant [Formula: see text]. Though widely expected, these properties are still unknown if the measure of [Formula: see text] is arbitrary.