THE DIRICHLET PROBLEM FOR MAXIMAL PLURISUBHARMONIC FUNCTIONS ON ANALYTIC VARIETIES IN ℂⁿ

Abstract

Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function [Formula: see text], we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on [Formula: see text].

Approximation of plurisubharmonic functions on complex varieties

“…Our final result generalizes Theorem 2.3 in [Wik2] in that it does not assume the existence of a B−regular neighborhood of V in C n . Theorem 1.6.…”

“…The theorem below deals with the problem of finding a bounded continuous maximal plurisubharmonic u on V such that the boundary values of u coincides with a given continuous function defined on part of the boundary ∂V. A weaker version of this result is given in Theorem 1.8 of [Wik2] under the assumption that V admits a B−regular neighborhood in C n . Recall that…”

“…This definition is taken from Definition 1.6 in [Wik2] and is analogous to the classical one given by Sadullaev for the case where V is an open set of C n (cf. Proposition 3.1.1 in [Kl]).…”

“…Before formulating it, recall that a complex variety V ⊂ D ⊂ C n is said to be locally irreducible if it is so at at every point a ∈ V i.e., there is a fundamental system of neighborhoods U j of a such that U j ∩V is irreducible in U j for every j. This point of local irreducibility was recorded incorrectly in Definition 1.1 of [Wik2] where it only requires a single neighborhood U of a such that U ∩V is locally irreducible. In general, it is quite hard to decide whether V is locally irreducible near its singular locus.…”

Plurisubharmonic approximation and boundary values of plurisubharmonic functions

“…Instead we wanted to show how Jensen measures can be used to study plurisubharmonic envelopes in situations without any affine structure. For a more thorough discussion on this topic, see Wikström [33].…”