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 Poincar\'e inequality, which yields quantitative stability estimates for the $L^p$-Poincar\'e 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 $$\lambda_{2,p}(I_D,V)-\lambda_{1,p}(I_D,V) \geq (p-1)(2^p-1)\left(\frac{\pi_p}{D}\right)^p$$ for every $p>1$ and every convex potential, with equality precisely for constant potentials.