Viscosity solutions to quaternionic Monge-Amp\`{e}re equations

TL;DR

This paper introduces a viscosity solution framework for quaternionic Monge-Ampère equations, providing comparison principles, solvability results, and establishing equivalence with pluripotential solutions, offering an alternative to traditional methods.

Contribution

It develops a viscosity approach for quaternionic Monge-Ampère equations, including comparison principles, existence results, and equivalence with pluripotential solutions.

Findings

Established a viscosity comparison principle.

Proved the existence of solutions to the Dirichlet problem.

Showed the equivalence between viscosity and pluripotential solutions.

Abstract

Quaternionic Monge-Amp\`{e}re 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-Amp\`{e}re equations $det(f)=F(q,f)$ with boundary value $f=g$ on $\partial\Omega$. Here $\Omega$ is a bounded domain on the quaternionic space $\mathbb{H}^n$, $g\in C(\partial\Omega)$, and $F(q,t)$ is a continuous function on $\Omega\times\mathbb{R}\rightarrow\mathbb{R}^+$ 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.