The plurisubharmonic Mergelyan property

• Main Author: Hed, Lisa
• Format: Doctoral Thesis
• Language: English
• Published: Umeå universitet, Institutionen för matematik och matematisk statistik 2012
• Subjects: Complex Monge-Ampère operator; approximation; plurisubharmonic function; subextension; Mergelyan property; plurisubharmonic function on compact sets; Jensen measures
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-52229; http://nbn-resolving.de/urn:isbn:978-91-7459-364-8

In this thesis, we study two different kinds of approximation of plurisubharmonic functions. The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside. The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.