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Global solutions and exterior Dirichlet problem for Monge-Ampère equation in \(\mathbb{R}^2\). (English) Zbl 1374.35188

Summary: 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.

MSC:

35J96 Monge-Ampère equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
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