Fundamental solutions of the complex Monge-Ampère equation

Halil Ibrahim Celik; Evgeny A. Poletsky

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Displaying similar documents to “Fundamental solutions of the complex Monge-Ampère equation”

A note on the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2010)

Annales Polonici Mathematici

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We prove the almost $^{1, 1}$ regularity of the degenerate complex Monge-Ampère equation in a special case.

Subextension of plurisubharmonic functions without changing the Monge-Ampère measures and applications

Le Mau Hai, Nguyen Xuan Hong (2014)

Annales Polonici Mathematici

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The aim of the paper is to investigate subextensions with boundary values of certain plurisubharmonic functions without changing the Monge-Ampère measures. From the results obtained, we deduce that if a given sequence is convergent in $C_{n - 1}$ -capacity then the sequence of the Monge-Ampère measures of subextensions is weakly*-convergent. As an application, we investigate the Dirichlet problem for a nonnegative measure μ in the class ℱ(Ω,g) without the assumption that μ vanishes on all pluripolar...

A regularity theorem for the complex Monge-Ampère equation in ℂPⁿ

Sławomir Kołodziej (2003)

Annales Polonici Mathematici

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$C^{1, 1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in $C^{1, 1}$ is proved.

On a Monge-Ampère type equation in the Cegrell class $_{χ}$

Rafał Czyż (2010)

Annales Polonici Mathematici

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Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation $- χ (u) (d d^{c} u) ⁿ = d μ$ . Under some additional assumption the solution u is uniquely determined.

A class of maximal plurisubharmonic functions

Azimbay Sadullaev (2012)

Annales Polonici Mathematici

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We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator $(d d^{c} e^{u}) ⁿ$ .

The complex Monge-Ampère equation for complex homogeneous functions in ℂⁿ

Rafał Czyż (2001)

Annales Polonici Mathematici

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We prove some existence results for the complex Monge-Ampère equation $(d d^{c} u) ⁿ = g d λ$ in ℂⁿ in a certain class of homogeneous functions in ℂⁿ, i.e. we show that for some nonnegative complex homogeneous functions g there exists a plurisubharmonic complex homogeneous solution u of the complex Monge-Ampère equation.

Pseudo-Riemannian and Hessian geometry related to Monge-Ampère structures

S. Hronek, R. Suchánek (2022)

Archivum Mathematicum

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We study properties of pseudo-Riemannian metrics corresponding to Monge-Ampère structures on four dimensional $T^{*} M$ . We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Plücker embedding equation. We also focus on pullbacks of the pseudo-metrics on two dimensional $M$ , and describe the corresponding Hessian structures.

On a problem of Monge.

Kantorovich, L.V. (2004)

Journal of Mathematical Sciences (New York)

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Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

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We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is $^{1, 1}$ and the right hand side has a density in $L^{p} (Ω)$ for some p > 1, and prove the Hölder continuity of the solution.

Subextension of plurisubharmonic functions without increasing the total Monge-Ampère mass

Rafał Czyż, Lisa Hed (2008)

Annales Polonici Mathematici

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We prove that subextension of certain plurisubharmonic functions is always possible without increasing the total Monge-Ampère mass.

On decomposable Monge-Ampère equations.

Machida, Y., Morimoto, T. (1999)

Lobachevskii Journal of Mathematics

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A characterization of bounded plurisubharmonic functions

Pham Hoang Hiep (2005)

Annales Polonici Mathematici

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We give a characterization for boundedness of plurisubharmonic functions in the Cegrell class ℱ.

On the Dirichlet problem in the Cegrell classes

Rafał Czyż, Per Åhag (2004)

Annales Polonici Mathematici

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Let μ be a non-negative measure with finite mass given by $φ (d d^{c} ψ) ⁿ$ , where ψ is a bounded plurisubharmonic function with zero boundary values and $φ \in L^{q} ((d d^{c} ψ) ⁿ)$ , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

Matrix inequalities and the complex Monge-Ampère operator

Jonas Wiklund (2004)

Annales Polonici Mathematici

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We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.

A counterexample to the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2005)

Annales Polonici Mathematici

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We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.

The Monge problem for strictly convex norms in $ℝ^{d}$

Thierry Champion, Luigi De Pascale (2010)

Journal of the European Mathematical Society

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We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of $ℝ^{d}$ under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.

The complex Monge-Ampère operator in the Cegrell classes

Rafał Czyż

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The complex Monge-Ampère operator is a useful tool not only within pluripotential theory, but also in algebraic geometry, dynamical systems and Kähler geometry. In this self-contained survey we present a unified theory of Cegrell's framework for the complex Monge-Ampère operator.

A unicity theorem for plurisubharmonic functions

Nguyen Quang Dieu (2011)

Annales Polonici Mathematici

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We give sufficient conditions for unicity of plurisubharmonic functions in Cegrell classes.

Solving the Degenerate Complex Monge-Ampère Equation with One Concentrated Singularity.

Laszló Lempert (1983)

Mathematische Annalen

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On formal theory of differential equations. II.

Jan Chrastina (1989)

Časopis pro pěstování matematiky

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Weak solutions to the complex Monge-Ampère equation on hyperconvex domains

Slimane Benelkourchi (2014)

Annales Polonici Mathematici

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We show a very general existence theorem for a complex Monge-Ampère type equation on hyperconvex domains.

A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the Monge-Ampère-type equation $det (A + λ g) = const .$ , where $A$ is the Schouten tensor of a conformally related metric and $λ > 0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique. ...