Shape optimisation of the maximum of the gradient of the torsion function on convex sets
Abstract
We analyse the boundary behaviour of the gradient of the torsion function wOm in convex domains Om⊂ℝd. We prove that its L∞-norm is attained at a boundary point and establish the stability of this norm under general (convex) perturbations. For problems of the form sup{|∇ wOm|∞: Om ⊂ Rd,textupopen, bounded and convex, G(Om)=m}, where G is either volume or perimeter, we show the existence of maximisers which are C1-smooth, rotationally symmetric about an axis and contain a (d−1)- dimensional ball (of estimated size) in their boundary. Other constraints, whether geometric (such as diameter or circumradius) or variational (such as torsion or first Dirichlet eigenvalue), are also briefly discussed, and some of their characteristic behaviours are highlighted.
Related
- Power convexity of the torsion function on horo-convex domains in hyperbolic space
- Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
- Concavity of the Ornstein--Uhlenbeck torsion function on centered ellipses and its failure on convex domains
- Optimality Conditions and Solution Set Characterisations of Convex Minimal Time Optimisation Problems
- Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation
BibTeX
@misc{berg2026shape,
title = {{Shape optimisation of the maximum of the gradient of the torsion function on convex sets}},
author = {Michiel van den Berg and Dorin Bucur and Mickael Nahon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08408},
eprint = {2610.08408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08408},
}
RIS
TY - UNPB
TI - Shape optimisation of the maximum of the gradient of the torsion function on convex sets
AU - Michiel van den Berg
AU - Dorin Bucur
AU - Mickael Nahon
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.08408
UR - https://arxiv.org/abs/2610.08408
AB - We analyse the boundary behaviour of the gradient of the torsion function w_Om in convex domains Om⊂ℝ^d. We prove that its L^∞-norm is attained at a boundary point and establish the stability of this norm under general (convex) perturbations. For problems of the form sup{|∇ w_Om|_∞: Om ⊂ R^d,textupopen, bounded and convex, G(Om)=m}, where G is either volume or perimeter, we show the existence of maximisers which are C^1-smooth, rotationally symmetric about an axis and contain a (d−1)- dimensional ball (of estimated size) in their boundary. Other constraints, whether geometric (such as diameter or circumradius) or variational (such as torsion or first Dirichlet eigenvalue), are also briefly discussed, and some of their characteristic behaviours are highlighted.
ER -