The obstacle problem for the Monge–Ampère equation with the lower obstacle

Abstract

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.