Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems

2021/11/02 by Mazzoleni, Dario, Pellacci, Benedetta, Verzini, Gianmaria · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2111.01491

Abstract

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain Ω⊂ ℝ^N, within a suitable class of sign-changing weights. 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 boundary of Ω and D is connected. Furthermore, the boundary of D intersects the boundary of the box Ω, and more precisely, \mathcal H^N-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.

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