Approximation of $m-$subharmonic functions on bounded domains in $\mathbb C^n$

TL;DR

This paper investigates how to approximate m-subharmonic functions on bounded domains in complex space using continuous m-subharmonic functions and explores boundary value problems for such functions.

Contribution

It introduces methods for approximating unbounded m-subharmonic functions and addresses boundary value problems with prescribed boundary data.

Findings

Approximation of unbounded m-subharmonic functions by continuous ones.

Existence results for boundary value problems with prescribed boundary data.

Techniques for handling small exceptional boundary subsets.

Abstract

Let $D$ be a bounded domain in $\mathbb C^n$. We study approximation of (not necessarily bounded from above) $m-$subharmonic function $D$ by continuous $m-$subharmonic ones defined on neighborhoods of $\overline{D}$. We also consider the existence of a $m-$subharmonic function on $D$ whose boundary values coincides with a given real valued continuous function on $\partial D$ except for a sufficiently small subset of $\partial D.$