A sharp quantitative nonlinear Poincaré inequality on convex domains

Abstract

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 into the lower bound an extra term depending on the second largest John semiaxis 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 asymptotically $\Omega$ 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 behavior of ${\mu }_p(\Omega ,\varphi )$ for varying weights and domains, including the case of collapsing geometries.