A Dirichlet problem for the complex Monge-Ampíre operator in F(f)

Åhag, Per

doi:10.1307/mmj/1177681988

Cited by 16 publications

(13 citation statements)

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“…Some particular cases of the classes E ( ) have been studied in [6,7,[9][10][11][12][13][14][15][16].…”

Section: International Journal Of Partial Differential Equationsmentioning

confidence: 99%

Weighted Pluricomplex Energy II

Benelkourchi

2015

International Journal of Partial Differential Equations

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We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes E by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure is the Monge-Ampère of a unique function ∈ E if and only if (E ) ⊂ 1 ( ). Then we show that if = ( ) for some ∈ E then = ( ) for some ∈ E , where is given boundary data. If moreover the nonnegative Borel measure is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.

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“…Some particular cases of the classes E ( ) have been studied in [6,7,[9][10][11][12][13][14][15][16].…”

Section: International Journal Of Partial Differential Equationsmentioning

confidence: 99%

Weighted Pluricomplex Energy II

Benelkourchi

2015

International Journal of Partial Differential Equations

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“…Some particular cases were studied in [1] and [8]. We do not know if every function in E has a boundary value, but we have the following theorem.…”

Section: The Boundary Valuesmentioning

confidence: 99%

“…The case f = 0 is Lemma 5.14 in [11] and the general case was proved in [1], where methods from [11] were used. The uniqueness can also be proved using Theorem 3.10 below.…”

Section: The Dirichlet Problem Inmentioning

confidence: 99%

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A general Dirichlet problem for the complex Monge–Ampère operator

Cegrell

2008

Ann. Polon. Math.

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Abstract. We study a general Dirichlet problem for the complex Monge-Ampère operator, with maximal plurisubharmonic functions as boundary data.1. Introduction. In classical potential theory, the Riesz representation theorem says that every negative subharmonic function can be written as a sum of a Green potential and a harmonic function. The smallest harmonic majorant of the potential is zero and the harmonic function is determined by its behaviour near the boundary. So it is natural to say that the harmonic part is the boundary value of the subharmonic function.The purpose of this paper is to formulate and study a pluripotential analogue of the classical setting.In pluripotential theory, the reminiscence of the Riesz representation theorem is inequality ( * ) below. Here the harmonic functions are replaced by the so called maximal functions, already considered by Bremermann in [6]. We refer to the books [15] and [16] for background and references.We will find that important results from classical potential theory carry over to our setting but we will also note some significant differences.In this paper we study a particular class of plurisubharmonic functions, the class E, defined in [11]. The reason for this choice is that the complex Monge-Ampère operator (dd c u) n is well-defined in this class and the maximal functions u in E are precisely the functions with (dd c u) n = 0.Subclasses of functions in E with continuous boundary values have been studied in [1] and [8]. The case of upper semicontinuous boundary values was considered in [14]. Here, we allow any bounded maximal function as boundary value when we show existence and uniqueness in the Dirichlet problem for the complex Monge-Ampère operator.

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“…Note that functions in E 0 (Ω) have zero boundary values; moreover, if u n ∈ F(Ω) then lim sup z →z u n (z ) = 0 for all z ∈ ∂Ω (cf. [1]). It might appear that (4.10) would suffice (without (4.9)) to prove Theorem 4.3.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%