Existence of entire solutions to the Monge-Ampère equation

Huaiyu Jian; Xu Jia Wang

doi:10.1353/ajm.2014.0029

Existence of entire solutions to the Monge-Ampère equation

Huaiyu Jian, Xu Jia Wang

Research output: Contribution to journal › Article › peer-review

19 Citations (Scopus)

Abstract

We prove the existence of infinitely many entire convex solutions to the Monge-Ampère equation det D²u = f in ℝⁿ, assuming that the inhomogeneous term f ≥ 0 and is of polynomial growth at infinity. When f satisfies the doubling condition, we show that solution is of polynomial growth. As an application, we resolve the existence of translating solutions to a class of Gauss curvature flow.

Original language	English
Pages (from-to)	1093-1106
Number of pages	14
Journal	American Journal of Mathematics
Volume	136
Issue number	4
DOIs	https://doi.org/10.1353/ajm.2014.0029
Publication status	Published - Aug 2014

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10.1353/ajm.2014.0029

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title = "Existence of entire solutions to the Monge-Amp{\`e}re equation",

abstract = "We prove the existence of infinitely many entire convex solutions to the Monge-Amp{\`e}re equation det D2u = f in ℝn, assuming that the inhomogeneous term f ≥ 0 and is of polynomial growth at infinity. When f satisfies the doubling condition, we show that solution is of polynomial growth. As an application, we resolve the existence of translating solutions to a class of Gauss curvature flow.",

author = "Huaiyu Jian and Wang, \{Xu Jia\}",

year = "2014",

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AU - Jian, Huaiyu

AU - Wang, Xu Jia

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AB - We prove the existence of infinitely many entire convex solutions to the Monge-Ampère equation det D2u = f in ℝn, assuming that the inhomogeneous term f ≥ 0 and is of polynomial growth at infinity. When f satisfies the doubling condition, we show that solution is of polynomial growth. As an application, we resolve the existence of translating solutions to a class of Gauss curvature flow.

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U2 - 10.1353/ajm.2014.0029

DO - 10.1353/ajm.2014.0029

M3 - Article

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VL - 136

SP - 1093

EP - 1106

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 4

ER -

Existence of entire solutions to the Monge-Ampère equation

Abstract

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