Monge–Ampère equation on exterior domains

By J. Bao, Haigang Li, Lei Zhang. 2013

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We consider the Monge–Ampère equation $$\det (D^2u)=f$$det(D2u)=f where $$f$$f is a positive function in $${\mathbb {R}}^n$$Rn and $$f=1+O(|x|^{-\beta })$$f=1+O(|x|-β) for some $$\beta >2$$β>2 at infinity. If the equation is globally defined on $${\mathbb {R}}^n$$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 $$n\ge 3$$n≥3 the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption $$\beta >2$$β>2 is sharp for all the results in this article.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7014f473bc5af3c2d5689aab3bec3e89ef3eaa02