2024/07/29 by Amato, Vincenzo, Bucur, Dorin, Fragalà, Ilaria
#35P15 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2407.20373
For any p ∈ ( 1, +∞), we give a new inequality for the first nontrivial Neumann eigenvalue μ_ p (Ω, φ) of the p-Laplacian on a convex domain Ω⊂ ℝN 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^ ∞ 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.