Uniqueness of least energy solutions for Monge–Ampère functional

Abstract

Let $\Omega$ be a bounded, smooth, uniformly convex domain in $\mathbb {R}^n$. We consider the following functional

$$\begin{aligned} \mathcal {E}[u]=\int _{\Omega }(-u)\det D^2 u dx,\quad \Vert u\Vert _{L^{q+1}(\Omega )}=1 \qquad \qquad (0.1) \end{aligned}$$

where $u\in C^2(\bar{\Omega })$ is convex and $u=0$ on $\partial \Omega$. In this paper, the uniqueness of least energy solution of (0.1) is investigated. For $n=2$, we prove the least energy solution of (0.1) is unique for $2<q<\infty$ provided it is locally uniformly convex. In particular, for $q=+\infty$, we show the uniqueness of the least energy solution of (0.1) and find its relation to Santalò point.