On hot spots of the torsion function and the ground state of the Laplacian
Abstract
We compare the maximum points PΩ and QΩ of the torsion function and the positive first Dirichlet eigenfunction. We begin with a systematic analytic description of both hot spots over the full range of opening angles of circular sectors: uniqueness, strict monotonicity with respect to the angle, distinct small-angle asymptotics, and explicit limiting equations in the slit-disk limit. We also prove the existence of a coincidence angle and establish arithmetic properties of the limiting coordinates. We then consider bounded convex planar domains with a line of symmetry, and we prove the sharp diameter-normalized identity supΩ (|PΩ−QΩ|)/diam(Ω)=1/2. Its upper-bound argument uses standard localization in the heart and an elementary geometric enclosure. Sharpness follows from thin convex domains with a vanishing relative width perturbation peaking near one endpoint: the torsion hot spot approaches that endpoint, whereas the ground-state hot spot approaches the midpoint.
Related
- Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
- Power convexity of the torsion function on horo-convex domains in hyperbolic space
- Shape optimisation of the maximum of the gradient of the torsion function on convex sets
- Tamagawa numbers and torsion of elliptic curves over function fields
- Concavity of the Ornstein--Uhlenbeck torsion function on centered ellipses and its failure on convex domains
BibTeX
@misc{he2026spots,
title = {{On hot spots of the torsion function and the ground state of the Laplacian}},
author = {Jiaxin He and Qinfeng Li and Shangfeng Wu and Shuangquan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41205},
eprint = {2610.41205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41205},
}
RIS
TY - UNPB
TI - On hot spots of the torsion function and the ground state of the Laplacian
AU - Jiaxin He
AU - Qinfeng Li
AU - Shangfeng Wu
AU - Shuangquan Xie
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41205
UR - https://arxiv.org/abs/2610.41205
AB - We compare the maximum points P_Ω and Q_Ω of the torsion function and the positive first Dirichlet eigenfunction. We begin with a systematic analytic description of both hot spots over the full range of opening angles of circular sectors: uniqueness, strict monotonicity with respect to the angle, distinct small-angle asymptotics, and explicit limiting equations in the slit-disk limit. We also prove the existence of a coincidence angle and establish arithmetic properties of the limiting coordinates. We then consider bounded convex planar domains with a line of symmetry, and we prove the sharp diameter-normalized identity sup_Ω (|P_Ω−Q_Ω|)/diam(Ω)=1/2. Its upper-bound argument uses standard localization in the heart and an elementary geometric enclosure. Sharpness follows from thin convex domains with a vanishing relative width perturbation peaking near one endpoint: the torsion hot spot approaches that endpoint, whereas the ground-state hot spot approaches the midpoint.
ER -