The complex Monge–Ampère operator

Czyż, Rafał

doi:10.4064/dm466-0-1

Cited by 21 publications

(15 citation statements)

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“…[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:…”

Section: Introductionmentioning

confidence: 99%

On Dirichlet's principle and problem

Åhag¹,

Cegrell²,

Czyż³

2012

MATH. SCAND.

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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.

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Section: Introductionmentioning

confidence: 99%

On Dirichlet's principle and problem

Åhag¹,

Cegrell²,

Czyż³

2012

MATH. SCAND.

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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%

“…Therefore the question of describing the measures which are the Monge-Ampère of bounded psh functions is very important for pluripotential theory, complex dynamic, and complex geometry. This problem has been studied extensively by various authors; see, for example, [2][3][4][5][6] and reference therein. In [7], Cegrell introduced the pluricomplex energy classes E (Ω) and F (Ω) ( ≥ 1) on which the complex Monge-Ampère operator is well defined.…”

Section: Introductionmentioning

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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“…Note that this also implies improved results in the pluricomplex case, m = n, and therefore, Theorem 6.3 also generalizes the stability result by Cegrell and Kołodziej [16]. For further information about these types of stability results in the case m = n, we refer to [19,Section 7.2].…”

Section: Introductionmentioning

confidence: 52%

“…In this section, we shall prove some new stability results for the complex Hessian operator. For previous results concerning stability of the complex Monge-Ampère equation or the complex Hessian equation, see, e.g., [16,19,34]. In those papers, the authors proved that under some assumption if μ j converges to μ, then the corresponding solutions U (μ j ) converges to U (μ) in capacity.…”

Section: Stability Of the Complex Hessian Operatormentioning

confidence: 99%

On a Family of Quasimetric Spaces in Generalized Potential Theory

Åhag

Czyż

2022

J Geom Anal

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We construct a family of quasimetric spaces in generalized potential theory containing m-subharmonic functions with finite (p, m)-energy. These quasimetric spaces will be viewed both in $${\mathbb {C}}^n$$ C n and in compact Kähler manifolds, and their convergence will be used to improve known stability results for the complex Hessian equations.

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The complex Monge–Ampère operator

Cited by 21 publications

References 61 publications

On Dirichlet's principle and problem

On Dirichlet's principle and problem

Weighted Pluricomplex Energy II

On a Family of Quasimetric Spaces in Generalized Potential Theory

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