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

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.