Existence of entire solutions to the Monge-Ampère equation
Existence of entire solutions to the Monge-Ampère equation
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DOI：
10.1353/ajm.2014.0029
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发表时间：
2014-07
影响因子：
1.7
通讯作者：
H. Jian;Xu-jia Wang
中科院分区：
文献类型：
--
作者：
H. Jian;Xu-jia Wang
本文证明了Monge-Schiere方程D^2 u=f$在${\Bbb R}^n$中的无穷多个整体凸解的存在性，假设非齐次项f\ge 0$和在无穷远处多项式增长.当f$满足加倍条件时，我们证明了解是多项式增长的.作为应用，我们解决了一类高斯曲率流平移解的存在性问题。
We prove the existence of infinitely many entire convex solutions to the Monge-Amp\`ere equation ${\rm det} D^2 u=f$ in ${\Bbb R}^n$, assuming that the inhomogeneous term $f\ge 0$ 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.