Existence, uniqueness and interior regularity of viscosity solutions for a class of Monge-Amp\`ere type equations

TL;DR

This paper establishes the existence, uniqueness, and interior regularity of viscosity solutions for a class of Monge-Ampère type equations on convex domains, addressing degenerate and singular boundary behaviors.

Contribution

It provides new results on the existence, uniqueness, and interior regularity of viscosity solutions for Monge-Ampère type equations, including degenerate and singular cases.

Findings

Existence and uniqueness of viscosity solutions

Interior regularity in various function spaces

Applicability to degenerate and singular boundary behaviors

Abstract

The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate the existence, uniqueness and interior regularity (including $W^{2,p}$ with $p\in(1,+\infty)$, $C^{1,\mu}$ with $\mu\in(0,1)$, and $C^\infty$) of the viscosity solution to the problem.