A quantitative convergence rate for torsional creep to the distance function as p tends to infinity
Abstract
We prove that the Dirichlet p-torsion function up converges to the distance to the boundary d at the rate O(1/p) in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for p>n and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals (1+log n)/p+O(p−2), so the order 1/p cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives up≤ KΩ1/(p−1)d for every p>1, with KΩ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of d is a supersolution whenever −Δd has a positive distributional lower bound. On C2 domains, the same barrier and the classical gradient maximum principle give norm∇ upL∞p−1≤ KΩ. When −Δd is a measure of finite total variation, which we prove for C1 domains with a uniform exterior ball, the gradients converge in Lq at the rate O(p−1/2) for q≤2 and O(p−1/q) for q≥2; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.
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BibTeX
@misc{bozorgnia2026quantitative,
title = {{A quantitative convergence rate for torsional creep to the distance function as p tends to infinity}},
author = {Farid Bozorgnia and Masoud Bayrami-Aminlouee and Khudoyor Mamayusupov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06534},
eprint = {2609.06534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A quantitative convergence rate for torsional creep to the distance function as p tends to infinity
AU - Farid Bozorgnia
AU - Masoud Bayrami-Aminlouee
AU - Khudoyor Mamayusupov
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2609.06534
UR - https://arxiv.org/abs/2609.06534
AB - We prove that the Dirichlet p-torsion function u_p converges to the distance to the boundary d at the rate O(1/p) in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for p>n and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals (1+log n)/p+O(p^−2), so the order 1/p cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives u_p≤ K_Ω^1/(p−1)d for every p>1, with K_Ω explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of d is a supersolution whenever −Δd has a positive distributional lower bound. On C^2 domains, the same barrier and the classical gradient maximum principle give norm∇ u_p_L^∞^p−1≤ K_Ω. When −Δd is a measure of finite total variation, which we prove for C^1 domains with a uniform exterior ball, the gradients converge in L^q at the rate O(p^−1/2) for q≤2 and O(p^−1/q) for q≥2; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.
ER -