Multidimensional Monge-Ampère Equation
Author | : A. V. Pogorelov |
Publisher | : |
Total Pages | : 103 |
Release | : 2008 |
Genre | : Monge-Ampère equations |
ISBN | : 9781904868811 |
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Author | : A. V. Pogorelov |
Publisher | : |
Total Pages | : 103 |
Release | : 2008 |
Genre | : Monge-Ampère equations |
ISBN | : 9781904868811 |
Author | : Alekseĭ Vasilʹevich Pogorelov |
Publisher | : |
Total Pages | : 103 |
Release | : 2008 |
Genre | : Monge-Ampère equations |
ISBN | : |
Author | : Nam Q. Le |
Publisher | : American Mathematical Society |
Total Pages | : 599 |
Release | : 2024-03-07 |
Genre | : Mathematics |
ISBN | : 1470474204 |
This book presents a systematic analysis of the Monge–Ampère equation, the linearized Monge–Ampère equation, and their applications, with emphasis on both interior and boundary theories. Starting from scratch, it gives an extensive survey of fundamental results, essential techniques, and intriguing phenomena in the solvability, geometry, and regularity of Monge–Ampère equations. It describes in depth diverse applications arising in geometry, fluid mechanics, meteorology, economics, and the calculus of variations. The modern treatment of boundary behaviors of solutions to Monge–Ampère equations, a very important topic of the theory, is thoroughly discussed. The book synthesizes many important recent advances, including Savin's boundary localization theorem, spectral theory, and interior and boundary regularity in Sobolev and Hölder spaces with optimal assumptions. It highlights geometric aspects of the theory and connections with adjacent research areas. This self-contained book provides the necessary background and techniques in convex geometry, real analysis, and partial differential equations, presents detailed proofs of all theorems, explains subtle constructions, and includes well over a hundred exercises. It can serve as an accessible text for graduate students as well as researchers interested in this subject.
Author | : Sławomir Kołodziej |
Publisher | : American Mathematical Soc. |
Total Pages | : 82 |
Release | : 2005 |
Genre | : Mathematics |
ISBN | : 082183763X |
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.
Author | : Sławomir Kołodziej |
Publisher | : American Mathematical Soc. |
Total Pages | : 64 |
Release | : 2005 |
Genre | : Mathematics |
ISBN | : 9781470404413 |
This is a collection of 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. Firstly 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.
Author | : Luis A. Caffarelli |
Publisher | : American Mathematical Soc. |
Total Pages | : 188 |
Release | : 1998-10-28 |
Genre | : Mathematics |
ISBN | : 9780821855621 |
In recent years, the Monge Ampere Equation has received attention for its role in several new areas of applied mathematics: As a new method of discretization for evolution equations of classical mechanics, such as the Euler equation, flow in porous media, Hele-Shaw flow, etc., As a simple model for optimal transportation and a div-curl decomposition with affine invariance and As a model for front formation in meteorology and optimal antenna design. These applications were addressed and important theoretical advances presented at a NSF-CBMS conference held at Florida Atlantic University (Boca Raton). L. Cafarelli and other distinguished specialists contributed high-quality research results and up-to-date developments in the field. This is a comprehensive volume outlining current directions in nonlinear analysis and its applications.
Author | : Cristian E. Gutiérrez |
Publisher | : Birkhäuser |
Total Pages | : 225 |
Release | : 2016-10-22 |
Genre | : Mathematics |
ISBN | : 3319433741 |
Now in its second edition, this monograph explores the Monge-Ampère equation and the latest advances in its study and applications. It provides an essentially self-contained systematic exposition of the theory of weak solutions, including regularity results by L. A. Caffarelli. The geometric aspects of this theory are stressed using techniques from harmonic analysis, such as covering lemmas and set decompositions. An effort is made to present complete proofs of all theorems, and examples and exercises are offered to further illustrate important concepts. Some of the topics considered include generalized solutions, non-divergence equations, cross sections, and convex solutions. New to this edition is a chapter on the linearized Monge-Ampère equation and a chapter on interior Hölder estimates for second derivatives. Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampère-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics. Both researchers and graduate students working on nonlinear differential equations and their applications will find this to be a useful and concise resource.
Author | : Luis A. Caffarelli |
Publisher | : American Mathematical Soc. |
Total Pages | : 186 |
Release | : 1999 |
Genre | : Mathematics |
ISBN | : 0821809172 |
In recent years, the Monge Ampère Equation has received attention for its role in several new areas of applied mathematics: as a new method of discretization for evolution equations of classical mechanics, such as the Euler equation, flow in porous media, Hele-Shaw flow, etc.; as a simple model for optimal transportation and a div-curl decomposition with affine invariance; and as a model for front formation in meteorology and optimal antenna design. These applications were addressed and important theoretical advances presented at a NSF-CBMS conference held at Florida Atlantic University (Boca Raton). L. Cafarelli and other distinguished specialists contributed high-quality research results and up-to-date developments in the field. This is a comprehensive volume outlining current directions in nonlinear analysis and its applications.
Author | : Roberto Moriyón |
Publisher | : |
Total Pages | : 102 |
Release | : 1979 |
Genre | : |
ISBN | : |
Author | : Luis A. Caffarelli |
Publisher | : |
Total Pages | : |
Release | : 1997 |
Genre | : |
ISBN | : |