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arXiv · 1502.07161

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

Jiguang Bao·Haigang Li·Lei Zhang
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Abstract

Monge-Amp\`ere 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-Amp\`ere 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.

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Jiguang Bao, Haigang Li, Lei Zhang. 2015-02-25. Global solutions and exterior Dirichlet problem for Monge-Ampere equation in $\mathbb R^2$. https://arxiv.org/abs/1502.07161

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