arXiv2024
For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $μ_ p (Ω, φ)$ of the $p$-Laplacian on a convex domain $Ω\subset \mathbb{R}^N$ with a power-concave weight $φ$. Our result improves the classical estimate in terms of the diameter, first stated in a seminal paper by Payne and Weinberger: we add in the lower bound an extra term depending on the second largest John semi-axis of $Ω$ (equivalent to a power of the width in the special case $N = 2$). The power exponent in the extra term is sharp, and the constant in front of it is explicitly tracked, thus enlightening the interplay between space dimension, nonlinearity and power-concavity. Moreover, we attack the stability question: we prove that, if $μ_ p (Ω, φ)$ is close to the lower bound, then $Ω$ is close to a thin cylinder, and $φ$ is close to a function which is constant along its axis. As intermediate results, we establish a sharp $L^ \infty$ estimate for the associated eigenfunctions, and we determine the asymptotic behaviour of $μ_ p (Ω, φ)$ for varying weights and domains, including the case of collapsing geometries.