An obstacle problem for a class of Monge-Amp\`ere type functionals

TL;DR

This paper investigates an obstacle problem related to Monge-Ampère type functionals, leading to fourth-order equations such as affine maximal surface equations and Abreu's equation, expanding understanding of these complex variational problems.

Contribution

It introduces a new obstacle problem framework for Monge-Ampère type functionals, connecting it to important fourth-order equations in differential geometry.

Findings

Establishes existence and regularity results for the obstacle problem.

Links the obstacle problem to affine maximal surface equations and Abreu's equation.

Provides new insights into the structure of solutions for these complex equations.

Abstract

In this paper we study an obstacle problem for Monge-Amp\`ere type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.