L∞ Variational Approximation of the Aubry Set
Abstract
Let H∈ C∞(ℝn×𝕋n) be a periodic Tonelli Hamiltonian with critical value c. For each k∈ℕ, let uk be the normalized minimizer of the variational functional introduced by Evans[7], Ik[w]=∫𝕋n ekH(Dw,x) dx, ∫𝕋nw dx=0. If u∞ is a uniform limit of a subsequence of {uk} and the Mather quotient (AM,δM) satisfies H1( AM,δM)=0, then u∞ is a critical subsolution that is strict outside A and A = {x∈𝕋n : Du∞(x) exists and H(Du∞(x),x)=c}={x∈𝕋n : u∞(x)=u−(x)}, where A is the projected Aubry set and u− is the backward weak KAM solution associated with u∞. In particular, by the theorem of Fathi--Figalli--Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on 𝕋n when n≤3. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between uk and its large-time backward Lax--Oleinik evolution.
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BibTeX
@misc{tran2026variational,
title = {{$L^\infty$ Variational Approximation of the Aubry Set}},
author = {Hung V. Tran and Yifeng Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01557},
eprint = {2609.01557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01557},
}
RIS
TY - UNPB
TI - L^∞ Variational Approximation of the Aubry Set
AU - Hung V. Tran
AU - Yifeng Yu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01557
UR - https://arxiv.org/abs/2609.01557
AB - Let H∈ C^∞(ℝ^n×𝕋^n) be a periodic Tonelli Hamiltonian with critical value c. For each k∈ℕ, let u_k be the normalized minimizer of the variational functional introduced by Evans[7], I_k[w]=∫_𝕋^n e^kH(Dw,x) dx, ∫_𝕋^nw dx=0. If u_∞ is a uniform limit of a subsequence of {u_k} and the Mather quotient (A_M,δ_M) satisfies H^1( A_M,δ_M)=0, then u_∞ is a critical subsolution that is strict outside A and A = {x∈𝕋^n : Du_∞(x) exists and H(Du_∞(x),x)=c}={x∈𝕋^n : u_∞(x)=u_−(x)}, where A is the projected Aubry set and u_− is the backward weak KAM solution associated with u_∞. In particular, by the theorem of Fathi--Figalli--Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on 𝕋^n when n≤3. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between u_k and its large-time backward Lax--Oleinik evolution.
ER -