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Quantitative homogenization of Monge-Ampère equations in periodic media
Abstract
Let uε and u be the convex solutions of det D2uε=F(x,x/ε), det D2u=F̅(x) on a bounded convex domain Ω, with the same Dirichlet data. Here, F is uniformly positive and periodic in its second variable, and F̅(x)=∫𝕋nF(x,y) dy. For m=0,1 and 0<α≤1, we prove |uε−u|L∞(Ω)≤ Cεm+α, ∀ 0<ε≤1, provided that u∈ Cm+2,α(Ω̅) and F∈ Cxm,α(Ω̅;Cy0,γ(𝕋n)) for some 0<γ<1. Both exponents are optimal in their respective regularity scales. We also establish the O(εα) rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.
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BibTeX
@misc{jin2026quantitative,
title = {{Quantitative homogenization of Monge-Ampère equations in periodic media}},
author = {Tianling Jin and YanYan Li and Hung V. Tran and Xushan Tu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03269},
eprint = {2610.03269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03269},
}
RIS
TY - UNPB
TI - Quantitative homogenization of Monge-Ampère equations in periodic media
AU - Tianling Jin
AU - YanYan Li
AU - Hung V. Tran
AU - Xushan Tu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03269
UR - https://arxiv.org/abs/2610.03269
AB - Let u^ε and u be the convex solutions of det D^2u^ε=F(x,x/ε), det D^2u=F̅(x) on a bounded convex domain Ω, with the same Dirichlet data. Here, F is uniformly positive and periodic in its second variable, and F̅(x)=∫_𝕋^nF(x,y) dy. For m=0,1 and 0<α≤1, we prove |u^ε−u|_L^∞(Ω)≤ Cε^m+α, ∀ 0<ε≤1, provided that u∈ C^m+2,α(Ω̅) and F∈ C_x^m,α(Ω̅;C_y^0,γ(𝕋^n)) for some 0<γ<1. Both exponents are optimal in their respective regularity scales. We also establish the O(ε^α) rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.
ER -