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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