Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains
Abstract
We study the fundamental gap of the Dirichlet p-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate Γp(ID,V)≥ (p−1)(2p−1)(πp/D)p, ID=(−D/2,D/2), for all 1<p<∞ and convex potentials V, with equality exactly for constant V. An analysis of collapsing convex domains shows that p=2 is exceptional: for p≠2, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For V≡0, define Gp,N:=infΩ⊂ℝNdiam(Ω)pΓp(Ω,0) over bounded convex domains. We prove G2,N=3π2 and Gp,N=0 for 1<p<2, N≥2, while for every p>2, Gp,N>0 and the infimum is attained. We obtain explicit dimension-dependent bounds and Gp,Nasympp Np−2 as N→∞, while for fixed N, limp↓2Gp,N=3π2 and limp→∞Gp,N1/p=4. We also characterize the collapse as p↓2: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals 2cos2(πt) dt on $(-1/2,1/2)$. The proof establishes log-concavity of the first p-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.
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BibTeX
@misc{chen2026fundamental,
title = {{Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains}},
author = {Rui Chen and Daniel Hauer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13443},
eprint = {2608.13443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13443},
}
RIS
TY - UNPB
TI - Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains
AU - Rui Chen
AU - Daniel Hauer
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.13443
UR - https://arxiv.org/abs/2608.13443
AB - We study the fundamental gap of the Dirichlet p-Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate Γ_p(I_D,V)≥ (p−1)(2^p−1)(π_p/D)^p, I_D=(−D/2,D/2), for all 1<p<∞ and convex potentials V, with equality exactly for constant V. An analysis of collapsing convex domains shows that p=2 is exceptional: for p≠2, the higher-dimensional gap is not governed by the sharp one-dimensional constant. For V≡0, define G_p,N:=inf_Ω⊂ℝ^Ndiam(Ω)^pΓ_p(Ω,0) over bounded convex domains. We prove G_2,N=3π^2 and G_p,N=0 for 1<p<2, N≥2, while for every p>2, G_p,N>0 and the infimum is attained. We obtain explicit dimension-dependent bounds and G_p,Nasymp_p N^p−2 as N→∞, while for fixed N, lim_p↓2G_p,N=3π^2 and lim_p→∞G_p,N^1/p=4. We also characterize the collapse as p↓2: minimizing domains degenerate, and every limiting normalized ground-state measure is supported on a unit segment and, up to rigid motions, equals 2cos^2(πt) dt on $(-1/2,1/2)$. The proof establishes log-concavity of the first p-eigenfunction with convex potential, a degenerate weighted Poincaré inequality for the ground-state measure, and a lower gap bound in terms of the first eigenvalue and the associated weighted Poincaré constant. These estimates also give the compactness and equality analysis for the collapsing minimizers.
ER -