Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains

Abstract

We study the fundamental gap of the Dirichlet $p$-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate ${\Gamma }_p(I_D,V)\ge (p-1)(2^p-1)({\pi }_p/D{)}^p$, $I_D=(-D/2,D/2)$, for all $1<p<\infty$ and convex potentials $V$, with equality exactly for constant $V$. An analysis of collapsing convex domains shows that $p=2$ is exceptional: for $p\ne 2$, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For $V\equiv 0$, define $G_{p,N}:={\inf }_{\Omega \subset {\mathbb{R}}^N}\mathrm{diam}(\Omega {)}^p{\Gamma }_p(\Omega ,0)$ over bounded convex domains. We prove $G_{2,N}=3{\pi }^2$ and $G_{p,N}=0$ for $1<p<2$, $N\ge 2$, while for every $p>2$, $G_{p,N}>0$ 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 $N$, ${\lim }_{p\downarrow 2}{G}_{p,N}=3{\pi }^2$ and ${\lim }_{p\to \infty }{G}_{p,N}^{1/p}=4$. We also characterize the collapse as $p\downarrow 2$: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals $2{\cos }^2(\pi t)dt$ 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.