Fundamental Gaps for the Dirichlet p-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy

Abstract

We study fundamental gaps for the Dirichlet \(p\)-Laplacian on bounded convex domains with convex potentials. In one dimension, we prove the sharp inequality \[ λ_{2,p}(I_D,V)-λ_{1,p}(I_D,V) \geq (p-1)(2^p-1)\left(\frac{π_p}{D}\right)^p \] for every \(p>1\) and every convex potential, with equality precisely for constant potentials. For \(N\geq2\), we identify a sharp transition at \(p=2\) through collapsing smooth convex domains: the gap vanishes for \(1<p<2\), remains of order \(D^{-2}\) for \(p=2\), and diverges for \(p>2\). In the regime \(p\geq2\), we prove log-concavity of the positive first eigenfunction by a regularization and two-point maximum principle. We then establish a degenerate weighted Poincaré inequality, which yields quantitative stability estimates for the \(L^p\)-Poincaré inequality and, in turn, quantitative lower bounds for the fundamental gap. For zero potential, we further obtain an enhanced gap estimate involving both the first eigenvalue and the diameter. Finally, we prove existence of diameter-normalized gap minimizers for \(p>2\) and show that they degenerate as \(p\downarrow2\), whereas for \(p=2\) the optimal gap is not attained by any bounded \(N\)-dimensional convex domain.