Approximation of Negative Plurisubharmonic Functions With Given Boundary Values

Abstract

In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Approximation of plurifinely plurisubharmonic functions

“…Therefore, it makes sense not only to ask for which domains Ω such an approximation is possible, but to ask for a characterization of those plurisubharmonic functions u that can be monotonically approximated from outside. According to the results by [5], [6], [9], [15] and the third author, this is possible if the domain Ω has the F -approximation property and u belongs to the Cegrell's classes in Ω.…”

On the approximation of weakly plurifinely plurisubharmonic functions

“…When Ω is bounded hyperconvex domain. According to the results by [4], [5], [8], [10] and other authors, the approximation is possible if the domain Ω has the F -approximation property and u belongs to one of the Cegrell's classes in Ω.…”

The locally F-approximation property of bounded hyperconvex domains

“…Some elements of pluripotential theory that will be used throughout the paper can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

“…Cegrell and Hed [6] proved in 2008 that a sufficient condition for Ω to have the F-approximation property is that one single function in the class N (Ω) can be approximated with functions in N (Ω j ). Hed [9] proved in 2010 that if Ω has the F-approximation property then we can approximate each function with given boundary values u ∈ F(Ω, f | Ω ) by an increasing sequence of functions u j ∈ F(Ω j , f | Ω j ) a.e. on Ω.…”