Volume 38, Issue 2
Monge-Ampère Equation with Bounded Periodic Data

Anal. Theory Appl., 38 (2022), pp. 128-147.

Published online: 2022-07

[An open-access article; the PDF is free to any online user.]

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• Abstract

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named author.

• Keywords

Monge-Ampère equation, Liouville theorem.

53C20, 53C21, 58J05, 35J60

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@Article{ATA-38-128, author = {Yanyan and Li and and 24091 and and Yanyan Li and Siyuan and Lu and and 24069 and and Siyuan Lu}, title = {Monge-Ampère Equation with Bounded Periodic Data}, journal = {Analysis in Theory and Applications}, year = {2022}, volume = {38}, number = {2}, pages = {128--147}, abstract = {

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named author.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0022}, url = {http://global-sci.org/intro/article_detail/ata/20796.html} }
TY - JOUR T1 - Monge-Ampère Equation with Bounded Periodic Data AU - Li , Yanyan AU - Lu , Siyuan JO - Analysis in Theory and Applications VL - 2 SP - 128 EP - 147 PY - 2022 DA - 2022/07 SN - 38 DO - http://doi.org/10.4208/ata.OA-0022 UR - https://global-sci.org/intro/article_detail/ata/20796.html KW - Monge-Ampère equation, Liouville theorem. AB -

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named author.

Yanyan Li & Siyuan Lu. (2022). Monge-Ampère Equation with Bounded Periodic Data. Analysis in Theory and Applications. 38 (2). 128-147. doi:10.4208/ata.OA-0022
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