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. We prove log-concavity of the positive first eigenfunction by a regularization and two-point maximum principle. 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). For (p\geq2) and convex potentials, we first establish a degenerate weighted Poincaré inequality, which yields quantitative stability estimates for the (L^p)-Poincaré inequality and, in turn, dimension-free bounds for the fundamental gap; for zero potential, we further obtain an enhanced gap estimate involving both the first eigenvalue and the diameter. We also prove existence of diameter-normalized gap minimizers for (p>2) and show that they degenerate as (p\downarrow2). Finally, for $N=1,$ 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.