Methods and Applications of Analysis

Volume 28 (2021)

Number 1

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