Monge-Ampere equation on exterior domains

2013/04/08 by Bao, Jiguang, Li, Haigang, Zhang, Lei · 2 citations
#35J67 #35J96 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.2415

Abstract

We consider the Monge-Ampère equation det(D²u)=f where f is a positive function in \mathbb Rⁿ and f=1+O(|x|^-β) for some β>2 at infinity. If the equation is globally defined on \mathbb Rⁿ we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for n≥ 3 the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption β>2 is sharp for all the results in this article.

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• Optimal asymptotic expansion of entire solutions to Monge-Ampère equation with C^α perturbed periodic data
• Exterior Dirichlet problem for Hessian equations on a non-convex ring

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