A comparison principle for parabolic complex Monge-Amp\`ere equations

TL;DR

This paper establishes a comparison principle for parabolic complex Monge-Ampère equations on pseudoconvex domains, facilitating the analysis of existence and uniqueness of viscosity solutions in complex analysis.

Contribution

It introduces a comparison principle for these equations and applies it to prove existence and uniqueness of viscosity solutions under specific conditions.

Findings

Proved a comparison principle for parabolic complex Monge-Ampère equations.

Demonstrated existence and uniqueness of viscosity solutions in certain cases.

Analyzed the behavior of solutions when zero sets are disjoint.

Abstract

In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on strongly pseudoconvex domains using the viscosity method. We prove a comparison principle for Parabolic complex Monge-Amp\`ere equations and use it to study the existence and uniqueness of viscosity solution in certain cases where the sets $\{z\in\Omega: f(t, z)=0 \}$ may be pairwise disjoint.