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

Szymon Pliś

Displaying similar documents to “A note on the regularity of the degenerate complex Monge-Ampère equation”

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.

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.

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 $(d d^{c} e^{u}) ⁿ$ .

Fundamental solutions of the complex Monge-Ampère equation

Halil Ibrahim Celik, Evgeny A. Poletsky (1997)

Annales Polonici Mathematici

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We prove that any positive function on ℂℙ¹ which is constant outside a countable $G_{δ}$ -set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.

On the Monge-Ampere differential equation $r t - s^{2} + λ^{2} (x, y) = 0$

B. N. Rachajsky (1969)

Matematički Vesnik

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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.

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.

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.

A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

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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. ...

A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $X$ be a compact Kähler manifold and $ω$ be a smooth closed form of bidegree $(1, 1)$ which is nonnegative and big. We study the classes $ℰ_{χ} (X, ω)$ of $ω$ -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class $ℰ_{χ} (X, ω)$ . This is done by...

A variational characterization of condenser capacities in $C^{n}$ within a class of plurisubharmonic functions

Jerzy Kalina (1983)

Annales Polonici Mathematici

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Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

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Let $(X, ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^{p}$ right hand side, $p > 1$ . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X, ω)$ of the complex Monge-Ampère operator acting on $ω$ -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^{p}$ -density belong to $(X, ω)$ and proving that...

Boundary values of functions in Cegrell’s class $_{ψ}$

Pham Hoang Hiep (2007)

Annales Polonici Mathematici

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We study boundary values of functions in Cegrell’s class $_{ψ}$ .

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...

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Concerning the energy class $_{p}$ for 0 < p < 1

Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)

Annales Polonici Mathematici

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The energy class $_{p}$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of $ℱ_{p}$ and its pluricomplex p-energy is proved.

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Annales Polonici Mathematici

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Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Henri Guenancia (2014)

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Annales Polonici Mathematici

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Sibel Şahin (2015)

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Poletsky-Stessin Hardy (PS-Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain Ω, the PS-Hardy space $H_{u}^{p} (Ω)$ is generated by a continuous, negative, plurisubharmonic exhaustion function u of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces $H_{u}^{p} (Ω)$ where the Monge-Ampère measure $(d d^{c} u) ⁿ$ has compact support for the associated...