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.

Disclosure

“t the development of this work. Conflict of interest. The authors declare that they have no conflict of interest. Data availability. No datasets were generated or analyzed during the current study. AI assistance statement. The authors used OpenAI models as assistive tools in preparing this manuscript. All mathematical arguments, proofs, and verifications were carried out by the authors, who take full responsibility for the content of the paper.”

PDF page 31

Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T05:13:51.472635Z

Structural counts