Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets
Abstract
Let H∈ C2(T*M) be a Tonelli Hamiltonian on a closed connected manifold and let uλ solve λuλ+H(x,Duλ)=c(H) in M. We study the convergence rate of uλ to the selected critical solution u0. Assume that the lifted Aubry set is a finite union Ã=Γ1sqcup⋯sqcupΓN, where each Γi is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove −Cλ≤ uλ−u0≤ Cλ|logλ|. Let μi be the projected Mather measure associated with Γi. If ∫M u0 dμi=0 for every i, then |uλ−u0|∞≤ Cλ. In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is O(λ). We give examples showing that both convergence rates O(λ) and O(λ|logλ|) are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order λ1/(2r−1) with r≥2. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.
Related
- Vanishing discount and viscosity selection problems for mechanical Hamilton--Jacobi equations
- Vanishing discount selection for constant-sign local perturbations of periodic Hamiltonians
- The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric
- Selection of critical solutions in non-monotone vanishing discount problems
- The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant
BibTeX
@misc{ni2026sharp,
title = {{Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets}},
author = {Panrui Ni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02779},
eprint = {2609.02779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02779},
}
RIS
TY - UNPB
TI - Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets
AU - Panrui Ni
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02779
UR - https://arxiv.org/abs/2609.02779
AB - Let H∈ C^2(T^*M) be a Tonelli Hamiltonian on a closed connected manifold and let u_λ solve λu_λ+H(x,Du_λ)=c(H) in M. We study the convergence rate of u_λ to the selected critical solution u_0. Assume that the lifted Aubry set is a finite union Ã=Γ_1sqcup⋯sqcupΓ_N, where each Γ_i is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove −Cλ≤ u_λ−u_0≤ Cλ|logλ|. Let μ_i be the projected Mather measure associated with Γ_i. If ∫_M u_0 dμ_i=0 for every i, then |u_λ−u_0|_∞≤ Cλ. In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is O(λ). We give examples showing that both convergence rates O(λ) and O(λ|logλ|) are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order λ^1/(2r−1) with r≥2. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.
ER -