Uniqueness for a system of Monge-Ampère equations

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 \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = γ|v|^p~&&\text{in} ~ Ω, \\\ \det D^2 v~& = μ|u|^{n^2/p}~&&\text{in} ~ Ω, \\\ u=v &= 0~&&\text{on}~ \partialΩ\end{alignedat} \right. \end{equation*} on bounded, smooth and uniformly convex domains $\Omega \subset R^n$ provided that $p$ is close to $n\ge 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $\Omega \subset R^n$.

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