Differential and Integral Equations

Global solutions and exterior Dirichlet problem for Monge-Ampère equation in $\mathbb R^2$

Jiguang Bao, Haigang Li, and Lei Zhang

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Abstract

Monge-Ampère equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in $\mathbb{R}^2$ or outside a convex set. First, we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.

Article information

Source
Differential Integral Equations Volume 29, Number 5/6 (2016), 563-582.

Dates
First available in Project Euclid: 9 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1457536891

Mathematical Reviews number (MathSciNet)
MR3471973

Subjects
Primary: 35J96: Elliptic Monge-Ampère equations 35J67: Boundary values of solutions to elliptic equations

Citation

Bao, Jiguang; Li, Haigang; Zhang, Lei. Global solutions and exterior Dirichlet problem for Monge-Ampère equation in $\mathbb R^2$. Differential Integral Equations 29 (2016), no. 5/6, 563--582. https://projecteuclid.org/euclid.die/1457536891.


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