Plurisubharmonic Approximation and Boundary Values of Plurisubharmonic Functions

TL;DR

This paper explores the approximation of plurisubharmonic functions on bounded domains, linking it to Dirichlet problems, introducing a stronger hyperconvexity notion, and characterizing boundary values for these functions.

Contribution

It introduces a stronger hyperconvexity concept, establishing a duality between approximation and Dirichlet problems for plurisubharmonic functions, and characterizes their boundary values.

Findings

Duality between approximation and Dirichlet problems is perfect for strongly hyperconvex domains.

Provides a characterization of boundary values of plurisubharmonic functions.

Proves new theorems on the approximation of plurisubharmonic functions.

Abstract

We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set $\bar\Omega$ in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.