Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction
Abstract
Let u>0 be the first Dirichlet p-eigenfunction on a bounded convex domain Ω⊂ℝN, and set XΩ:=|∇log u|p−2∇log u. We study whether the sharp one-dimensional two-point modulus for XΩ, which for p=2 reduces to the logarithmic-gradient estimate of Andrews and Clutterbuck, persists for p≠2. We prove sharp estimates on intervals and balls for every p>1, with the radial modulus on balls strictly larger than the one-dimensional one. In dimensions N≥2, however, the one-dimensional modulus fails on general convex domains for every p≠2. For 1<p<2, at every sufficiently small fixed scale there are smooth uniformly convex domains for which the corresponding two-point flux tends to zero. For p>2, on thin domains Ωε=D×(−ε,ε), with D⊂ℝN−1 bounded and convex, the normalized first eigenfunctions satisfy uε(x,ε z)⟶ ϕ(z)/ϕ(0) (GD(x)/GD(x0))2/p locally uniformly in D×(−1,1), where ϕ and GD are the first Dirichlet eigenfunctions of the p-Laplacian on (−1,1) and of the Laplacian on D, respectively. This yields the failure for p>2. Finally, for arbitrary C2 functions, positivity of the symmetric differential of the p-gradient implies convexity, and the converse holds universally if and only if p=2.
Related
- Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space
- Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator
- A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian
- Endpoint eigenfunction restriction estimates in codimension two
- Eigenfunction asymptotics and nodal domain estimates for dumbbell domains
BibTeX
@misc{chen2026twopoint,
title = {{Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction}},
author = {Rui Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29195},
eprint = {2608.29195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29195},
}
RIS
TY - UNPB
TI - Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction
AU - Rui Chen
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.29195
UR - https://arxiv.org/abs/2608.29195
AB - Let u>0 be the first Dirichlet p-eigenfunction on a bounded convex domain Ω⊂ℝ^N, and set X_Ω:=|∇log u|^p−2∇log u. We study whether the sharp one-dimensional two-point modulus for X_Ω, which for p=2 reduces to the logarithmic-gradient estimate of Andrews and Clutterbuck, persists for p≠2. We prove sharp estimates on intervals and balls for every p>1, with the radial modulus on balls strictly larger than the one-dimensional one. In dimensions N≥2, however, the one-dimensional modulus fails on general convex domains for every p≠2. For 1<p<2, at every sufficiently small fixed scale there are smooth uniformly convex domains for which the corresponding two-point flux tends to zero. For p>2, on thin domains Ω_ε=D×(−ε,ε), with D⊂ℝ^N−1 bounded and convex, the normalized first eigenfunctions satisfy u_ε(x,ε z)⟶ ϕ(z)/ϕ(0) (G_D(x)/G_D(x_0))^2/p locally uniformly in D×(−1,1), where ϕ and G_D are the first Dirichlet eigenfunctions of the p-Laplacian on (−1,1) and of the Laplacian on D, respectively. This yields the failure for p>2. Finally, for arbitrary C^2 functions, positivity of the symmetric differential of the p-gradient implies convexity, and the converse holds universally if and only if p=2.
ER -