---
abstract: 'In this paper we study an obstacle problem for Monge-Ampère type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu’s equation.'
address:
- |
    Department of Mathematics\
    Princeton University\
    Fine Hall, Washington Road\
    Princeton, NJ 08544-1000, USA.
- |
    School of Mathematical Sciences and Beijing International Center for Mathematical Research\
    Peking University\
    Beijing 100871, China.
author:
- Jiakun Liu
- Bin Zhou
title: 'An obstacle problem for a class of Monge-Ampère type functionals'
---

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=16.4pt

Introduction {#s1}
============

Free boundary and obstacle problems for partial differential equations have been studied extensively in the past decades. For Monge-Ampère equations, obstacle problems were studied in [@CW; @Lee; @Sa] among others, and a related free boundary problem was studied in [@CM]. In this paper we consider an obstacle problem for the functional $$\label{e101}
   J_\alpha(u)=\begin{cases}
                       \int_\Omega \left[{\mbox{det}}\,D^2u\right]^{\alpha}
                       -\alpha\int_\Omega fu, &\ \alpha>0 \ 
\text{and}\ \alpha\neq 1, \\[5pt]
               \int_\Omega \log{\mbox{det}}\,D^2u -\int_\Omega fu, &\ \alpha=0,
                      \end{cases}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $f\in L^\infty(\Omega)$. For simplicity, we denote the nonlinear part of the functional by $A_\alpha(u)$, see . We would like to study the maximization problem $$\label{e102}
   J_\alpha(u)=\sup\left\{J_\alpha(v)\,:\,v\in\mathcal{S}[\varphi,\psi]\right\},$$ where $\mathcal{S}[\varphi,\psi]$ 