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

TL;DR

This paper investigates the asymptotic behavior of the optimal favorable region in a Neumann spectral problem, revealing its shape, location, and relation to boundary curvature as the region's measure diminishes.

Contribution

It provides a detailed asymptotic analysis of the optimal region's shape and position in a weighted Neumann eigenvalue minimization problem, addressing longstanding questions.

Findings

Maximum point of eigenfunction approaches boundary point with maximal mean curvature.

Optimal region becomes nearly spherical with quantifiable asymmetry decay.

Shape and location of the optimal set are characterized in the small volume limit.

Abstract

We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $\Omega\subset \mathbb{R}^{N}$, $N\ge2$, 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 $u$, $P\in \partial \Omega$, tends to a point of maximal mean curvature of $\partial \Omega$. Furthermore, we show that $D$ is the intersection with $\Omega$ 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.