The Complex Monge-Ampere Equation and Pluripotential Theory

The Complex Monge-Ampere Equation and Pluripotential Theory

Author: Sławomir Kołodziej

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 82

ISBN-13: 082183763X

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We collect here results on the existence and stability of weak solutions of complex Monge-Ampere equation proved by applying pluripotential theory methods and obtained in past three decades. First we set the stage introducing basic concepts and theorems of pluripotential theory. Then the Dirichlet problem for the complex Monge-Ampere equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part, the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampere equation on compact Kahler manifolds. This is a generalization of the Calabi-Yau theorem.


Algebra, Complex Analysis, and Pluripotential Theory

Algebra, Complex Analysis, and Pluripotential Theory

Author: Zair Ibragimov

Publisher: Springer

Published: 2018-10-11

Total Pages: 224

ISBN-13: 3030011445

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This book features papers presented during a special session on algebra, functional analysis, complex analysis, and pluripotential theory. Research articles focus on topics such as slow convergence, spectral expansion, holomorphic extension, m-subharmonic functions, pseudo-Galilean group, involutive algebra, Log-integrable measurable functions, Gibbs measures, harmonic and analytic functions, local automorphisms, Lie algebras, and Leibniz algebras. Many of the papers address the theory of harmonic functions, and the book includes a number of extensive survey papers. Graduate and researchers interested in functional analysis, complex analysis, operator algebras and non-associative algebras will find this book relevant to their studies. The special session was part of the second USA-Uzbekistan Conference on Analysis and Mathematical Physics held on August 8-12, 2017 at Urgench State University (Uzbekistan). The conference encouraged communication and future collaboration among U.S. mathematicians and their counterparts in Uzbekistan and other countries. Main themes included algebra and functional analysis, dynamical systems, mathematical physics and partial differential equations, probability theory and mathematical statistics, and pluripotential theory. A number of significant, recently established results were disseminated at the conference’s scheduled plenary talks, while invited talks presented a broad spectrum of findings in several sessions. Based on a different session from the conference, Differential Equations and Dynamical Systems is also published in the Springer Proceedings in Mathematics & Statistics Series.


Pluripotential Theory

Pluripotential Theory

Author: Giorgio Patrizio

Publisher: Springer

Published: 2013-05-16

Total Pages: 328

ISBN-13: 3642364217

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Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This volume brings together the lectures held at the 2011 CIME session on "pluripotential theory" in Cetraro, Italy. This CIME course focused on complex Monge-Ampére equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.


Nonlinear Potential Theory and Weighted Sobolev Spaces

Nonlinear Potential Theory and Weighted Sobolev Spaces

Author: Bengt O. Turesson

Publisher: Springer

Published: 2007-05-06

Total Pages: 188

ISBN-13: 3540451684

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The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.


Pluripotential Theory

Pluripotential Theory

Author: Maciej Klimek

Publisher:

Published: 2023

Total Pages: 0

ISBN-13: 9781383025705

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This monograph presents a recently developed branch of analysis in several complex variables. It focuses on the interplay between maximal plurisubharmonic functions and the generalized complex Monge-Ampere operator.


Selected Questions of Mathematical Physics and Analysis

Selected Questions of Mathematical Physics and Analysis

Author: I. V. Volovich

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 420

ISBN-13: 9780821804643

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This collection, dedicated to the 70th anniversary of the birth of VasiliiSergeevich Vladimirov, consists of original papers on various branches of analysis and mathematical physics. It presents work relating to the following topics:--the theory of generalized functions--complex and $p$-adic analysis--mathematical questions of quantum field theory and statistical mechanics--computational mathematics and differential equations.


Pluripotential Theory

Pluripotential Theory

Author: Maciej Klimek

Publisher:

Published: 1991

Total Pages: 296

ISBN-13:

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Pluripotential theory is a recently developed non-linear complex counterpart of classical potential theory. Its main area of application is multidimensional complex analysis. The central part of the pluripotential theory is occupied by maximal plurisubharmonic functions and the generalized complex Monge-Ampere operator. The interplay between these two concepts provides the focal point of this monograph, which contains an up-to-date account of the developments from the large volume of recent work in this area. A substantial proportion of the work is devoted to classical properties of subharmonic and plurisubharmonic functions, which makes the pluripotential theory available for the first time to a wide audience of analysts.