Subextension and approximation of negative plurisubharmonic functions

Abstract

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampere mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampere mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampere mass, then we can find a plurisubharmonic function u defined on Ω_2, with given boundary values, such that u <= u on Ω and with control over the Monge-Ampere mass of u.