Viscosity solutions to quaternionic Monge-Ampère equations

Abstract

Quaternionic Monge-Ampère equations have recently been studied intensively using methods from pluripotential theory. We present an alternative approach by using the viscosity methods. We study the viscosity solutions to the Dirichlet problem for quaternionic Monge-Ampère equations $det(f)=F(q,f)$ with boundary value $f=g$ on $\partialΩ$. Here $Ω$ is a bounded domain on the quaternionic space $\mathbb{H}^n$, $g\in C(\partialΩ)$, and $F(q,t)$ is a continuous function on $Ω\times\mathbb{R}\rightarrow\mathbb{R}^+$ which is non-decreasing in the second variable. We prove a viscosity comparison principle and a solvability theorem. Moreover, the equivalence between viscosity and pluripotential solutions is showed.