*Published Paper*

**Inserted:** 2 nov 2021

**Last Updated:** 11 oct 2023

**Journal:** SIAM J. Math. Anal.

**Year:** 2021

**Abstract:**

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$, 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 $\Omega$ and $D$ is connected.
Furthermore, the boundary of $D$ intersects the boundary of the box $\Omega$, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial \Omega)\ge 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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