Anal. Theory Appl., 38 (2022), pp. 128-147.
Published online: 2022-07
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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} }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.