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 $N\ge 2$, 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 $p\ge 2$ 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 $p\downarrow 2$. Finally, for $N=1,$ we prove the sharp inequality $${\lambda }_{2,p}(I_D,V)-{\lambda }_{1,p}(I_D,V)\ge (p-1)(2^p-1){\left(\frac{{\pi }_p}{D}\right)}^p$$ for every $p>1$ and every convex potential, with equality precisely for constant potentials.