We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain Ω ⊂ RN, N ≥ 2, within a suit- able class of sign-changing weights. This problem arises in the study of the persistence of a species in population dynamics. Denoting with u the optimal eigenfunction and with D its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of D tends to zero. We show that, when the measure of D is sufficiently small, u has a unique local maximum point lying on the bound- ary of Ω and D is connected. Furthermore, the boundary of D intersects the boundary of the box Ω, and more precisely, HN−1(∂D ∩ ∂Ω) ≥ C|D|(N−1)/N for some universal constant C > 0. Though widely expected, these properties are still unknown if the measure of D is arbitrary.
Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems
Benedetta Pellacci;
2023
Abstract
We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain Ω ⊂ RN, N ≥ 2, within a suit- able class of sign-changing weights. This problem arises in the study of the persistence of a species in population dynamics. Denoting with u the optimal eigenfunction and with D its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of D tends to zero. We show that, when the measure of D is sufficiently small, u has a unique local maximum point lying on the bound- ary of Ω and D is connected. Furthermore, the boundary of D intersects the boundary of the box Ω, and more precisely, HN−1(∂D ∩ ∂Ω) ≥ C|D|(N−1)/N for some universal constant C > 0. Though widely expected, these properties are still unknown if the measure of D is arbitrary.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
