On Dirichlet's principle and problem

Abstract

Abstract. The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampère operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods. IntroductionThroughout this note let Ω ⊆ C n , n ≥ 1, be a bounded, connected, open, and hyperconvex set. By E 0 we denote the family of all bounded plurisubharmonic functions ϕ defined on Ω such that lim z→ξ ϕ(z) = 0 for every ξ ∈ ∂Ω , andwhere (dd c · ) n is the complex Monge-Ampère operator. Next let E p , p > 0, denote the family of plurisubharmonic functions u defined on Ω such that there exists a decreasing sequence {u j }, u j ∈ E 0 , that converges pointwise to u on Ω, as j tends to +∞, and 10,14]). It should be noted that it follows from [10] that the complex Monge-Ampère operator is well-defined on E p . It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,17]). For further information on pluripotential theory we refer to [16,19,20]. The purpose of this paper is to give a new proof of Theorem B below and use Theorem B to prove (2) implies (1) the following theorem:Theorem A (Dirichlet's problem). Let µ be a non-negative Radon measure, then the following conditions are equivalent:(1) there exists a function u ∈ E 1 such that (dd c u) n = µ, (2) there exists a constant B > 0, such that Ω (−ϕ) dµ ≤ B e 1 (ϕ) 1 n+1 for all ϕ ∈ E 1 , (1.1) (3) the class E 1 is contained in L 1 (µ),2000 Mathematics Subject Classification. Primary 35J20; Secondary 32W20.

Shakedown

“…To prove this result we use a variational method introduced in [5]. Our result generalizes the result in [2]. Using this and following [8] we also get: Theorem 2.…”

A variational approach to complex Hessian equations in Cn

“…The class F a m (Ω) consists of non-positive m-subharmonic functions whose Hessian measures are well-defined, of finite total mass and do not charge m-polar sets. In Section 4, we develop a variational approach inspired by [5] (see also [2]) to solve equation (1.1) with a "finite energy" right-hand side.…”

“…These classes were proven to be a fruitful branch of research in the recent years. A variational approach to the Dirichlet problem on a hyperconvex domain of C n (which had been studied by Cegrell [8,9]) was recently done in [2].…”

“…We characterize the range of H m (u) when u runs in E p m (Ω). Our results are direct generalizations of the classical case of plurisubharmonic functions (see [2,8,9]). The variational approach for the complex Monge-Ampère equation was first introduced in [5].…”

Weighted Pluricomplex Energy II

“…where E 0 (Ω) is the cone of all bounded psh functions defined on the domain Ω with finite total Monge-Ampère mass and lim → ( ) = 0, for every ∈ Ω. Recently,Åhag et al in [8] proved that, in the case = 1, inequality (1) is equivalent to E 1 (Ω) ⊂ 1 ( ). In this note, our first objective is to extend this result by showing that, for all positive number , inequality (1) is equivalent to E (Ω) ⊂ ( ).…”