In this paper, we study the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the Monge–Ampère equation with the lower obstacle function, which arises in the prescribed Gauss curvature with an obstacle.

The main feature of this paper is that we consider the obstacle problem for Monge–Ampère operator, which is a log-concave operator, with the lower obstacle function. Generally, the problem for a convex operator with a lower obstacle or a concave operator with an upper obstacle is considered due to the definition of the solutions and the classification of the global solutions. In this paper, difficulties caused by the incongruousness of the operator and the location of the obstacle are considered in many parts such as the penalization problem, classification of the global solutions, and the directional monotonicity.

在本文中，我们研究具有较低障碍函数的Monge-Ampère方程的障碍问题的解的存在性和最佳正则性以及障碍问题的自由边界的正则性，该方程是在具有障碍的规定高斯曲率下产生的。

本文的主要特点是，我们考虑了具有较低障碍物功能的对数凹面算子Monge–Ampère算子的障碍问题。通常，由于解的定义和整体解的分类，考虑了具有较低障碍的凸算子或具有较高障碍的凹算子的问题。在本文中，许多方面都考虑了由于操作员的不协调和障碍物的位置而造成的困难，例如惩罚问题，整体解的分类以及方向单调性。