Weak solution of Parabolic complex Monge-Amp\'ere equation II

TL;DR

This paper investigates the existence and properties of weak solutions to the parabolic complex Monge-Ampère equation within a bounded, strictly pseudoconvex domain in complex space, focusing on smooth boundary data and arbitrary initial plurisubharmonic functions.

Contribution

It extends the theory of weak solutions for the parabolic complex Monge-Ampère equation to more general initial conditions in a strictly pseudoconvex domain.

Findings

Existence of weak solutions under specified conditions

Analysis of boundary and initial value behavior

Extension of previous results to broader initial data

Abstract

We study the Parabolic complex Monge-Amp\'ere equation in a bounded strictly pseudoconvex domain in \mathbb{C}^n, with the boundary condition u=\varphi and the initial condition u=u_0. In this paper, we consider the case where \varphi is smooth and u_0 is an arbitrary plurisubharmonic function in a neighbourhood of \Omega satisfying u_0=\varphi(.,0) on the boundary of \Omega.