Uniqueness for a system of Monge-Ampère equations

2020/01/07 by Nam Q. Le, Le, Nam Q.
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2001.02196

v2: minor typos fixed; to appear in Methods Appl. Anal.'s special volume for John Urbas' 60th birthday

arxiv created 2020/06/11 · arxiv updated 2020/06/12

Abstract

In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Ampère equations \ \beginalignedat2 det D² u~ = γ|v|^p~&in ~ Ω,
det D² v~ = μ|u|^n2/p~&in ~ Ω,
u=v = 0~&on~ ∂Ω\endalignedat . on bounded, smooth and uniformly convex domains Ω⊂ Rⁿ provided that p is close to n≥ 2. When p=n, we show that the uniqueness holds for general bounded convex domains Ω⊂ Rⁿ.

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