# Methods and Applications of Analysis

## Volume 28 (2021)

### Special Issue for the 60th Birthday of John Urbas: Part II

Guest Editors: Neil Trudinger and Xu-Jia Wang

### Uniqueness for a system of Monge–Ampère equations

Pages: 15 – 30

DOI:  https://dx.doi.org/10.4310/MAA.2021.v28.n1.a2

#### Author

Nam Q. Le (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

#### 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{cases}\operatorname {det} D^2 u = \gamma \lvert v \rvert p & \textrm{in} \; \Omega , \\\operatorname {det} D^2 v = \mu {\lvert u \rvert }^{n^ 2 / p} & \textrm{in} \; \Omega , \\\quad u = v = 0 & \textrm{on} \; \partial \Omega\end{cases}on bounded, smooth and uniformly convex domains $\Omega \subset \mathbb{R}^n$ provided that $p$ is close to $n \geq 2$. When $p = n$, we show that the uniqueness holds for general bounded convex domains $\Omega \subset \mathbb{R}^n$.

#### Keywords

system of Monge–Ampère equations, uniqueness, eigenvalue problem

#### 2010 Mathematics Subject Classification

35A02, 35J70, 35J96, 47A75

The research of the author was supported in part by the National Science Foundation under grant DMS-1764248.

Received 11 December 2019

Accepted 9 June 2020

Published 1 December 2021