On the Monge-Ampere differential equation $rt - s^{2} + \lambda ^{2}(x,y)=0$

B. N. Rachajsky

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

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...

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 $_{ψ}$ .

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.

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

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We show that if a decreasing sequence of subharmonic functions converges to a function in $W_{l o c}^{1, 2}$ then the convergence is in $W_{l o c}^{1, 2}$ .

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

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...

Addendum to “On the $p^{λ}$ problem”

Stephan Baier (2005)

Acta Arithmetica

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On the fundamental solution of the equation $∆^{p} u (X) + k (r) u (X) = 0$

J. Musiałek (1969)

Annales Polonici Mathematici

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Green functions, Segre numbers, and King’s formula

Mats Andersson, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$ , and let $G$ be a plurisubharmonic function such that $G = log | f | + 𝒪 (1)$ locally at $Z$ , where $f$ is a tuple of holomorphic functions that defines $𝒥$ . We give a meaning to the Monge-Ampère products ${(d d^{c} G)}^{k}$ for $k = 0, 1, 2, ...$ , and prove that the Lelong numbers of the currents $M_{k}^{𝒥} : = 1_{Z} {(d d^{c} G)}^{k}$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$ , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_{k}^{𝒥}$ satisfy a certain...