Year: 2026
Type: preprint
Abstract: We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate $Γ_p(I_D,V)\ge (p-1)(2^p-1)(π_p/D)^p$, $I_D=(-D/2,D/2)$, for all $12$, $G_{p,N}>0$ and the infimum is attained. We obtain explicit dimension-dependent bounds and $G_{p,N}\asymp_p N^{p-2}$ as $N\to\infty$, while for fixed $N$, $\lim_{p\downarrow2}G_{p,N}=3π^2$ and $\lim_{p\to\infty}G_{p,N}^{1/p}=4$. We also characterize the collapse as $p\downarrow2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2\cos^2(πt)\,dt$ on $(-1/2,1/2)$. The proof establishes log-concavity of the first $p$-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.
Source: arXiv (Cornell University)
Authors: Chen, Rui, Hauer, Daniel
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203487615