A sharp quantitative nonlinear Poincar\'e inequality on convex domains

For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $\mu _ p (\Omega, \varphi)$ of the $p$-Laplacian on a convex domain $\Omega \subset \mathbb{R}^N$ with a power-concave weight $\varphi$. 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 $\Omega$ (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 $\mu _ p (\Omega, \varphi)$ is close to the lower bound, then $\Omega$ is close to a thin cylinder, and $\varphi$ 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 $\mu _ p (\Omega, \varphi)$ for varying weights and domains, including the case of collapsing geometries.