Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem

We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain Ω ⊂ R N , N ≥ 2 , for the weight varying in a suitable class of sign-changing bounded functions. Denoting with u the optimal eigenfunction and with D its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of D tends to zero, the unique maximum point of u , P ∈ ∂ Ω , tends to a point of maximal mean curvature of ∂Ω. Furthermore, we show that D is the intersection with Ω of a C 1 , 1 nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of D . These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.