Viscosity solutions to parabolic complex Monge-Amp\`ere equations

TL;DR

This paper develops a viscosity approach to solve parabolic complex Monge-Ampère equations on pseudoconvex domains, proving existence, convergence, and regularity of solutions.

Contribution

It extends previous results by establishing existence, convergence at infinity, and Hölder regularity for solutions using viscosity methods.

Findings

Proved existence of viscosity solutions for the equations.

Established convergence of solutions at infinity.

Demonstrated Hölder regularity of solutions with Hölder continuous data.

Abstract

In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on a strongly pseudoconvex domain by the viscosity method. We extend the results in [EGZ15b] on the existence of solution and the convergence at infinity. We also establish the H\"older regularity of the solutions when the Cauchy-Dirichlet data are H\"older continuous.