Approximation and Subextension 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 ? in Cn and has essentially boundary values zero and bounded Monge-Ampère 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-Ampère 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 ?_1 and ?_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on ?_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on ?_2, with given boundary values, such that û <= u on ? and with control over the Monge-Ampère mass of û.