Nonlinear Analysis on Manifolds. Monge-Ampère Equations
Nonlinear Analysis on Manifolds. Monge-Ampère Equations

Author: Thierry Aubin

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 215

ISBN-13: 1461257344

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This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.

Analysis of Monge–Ampère Equations
Analysis of Monge–Ampère Equations

Author: Nam Q. Le

Publisher: American Mathematical Society

Published: 2024-03-08

Total Pages: 599

ISBN-13: 1470476258

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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.

Sheaves on Manifolds
Sheaves on Manifolds

Author: Masaki Kashiwara

Publisher: Springer Science & Business Media

Published: 2002-05-01

Total Pages: 536

ISBN-13: 9783540518617

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Sheaf Theory is modern, active field of mathematics at the intersection of algebraic topology, algebraic geometry and partial differential equations. This volume offers a comprehensive and self-contained treatment of Sheaf Theory from the basis up, with emphasis on the microlocal point of view. From the reviews: "Clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics." –Bulletin of the L.M.S.

Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities

Author: Emmanuel Hebey

Publisher: American Mathematical Soc.

Published: 2000-10-27

Total Pages: 306

ISBN-13: 0821827006

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This volume offers an expanded version of lectures given at the Courant Institute on the theory of Sobolev spaces on Riemannian manifolds. ``Several surprising phenomena appear when studying Sobolev spaces on manifolds,'' according to the author. ``Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a central role.'' The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds. Chapter 3 presents the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5. Chapter 6 presents special types of Sobolev inequalities under constraints. Best constants problems for complete noncompact manifolds are discussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. And Chapter 9 discusses the influence of symmetries on Sobolev embeddings. An appendix offers brief notes on the case of manifolds with boundaries. This topic is a field undergoing great development at this time. However, several important questions remain open. So a substantial part of the book is devoted to the concept of best constants, which appeared to be crucial for solving limiting cases of some classes of PDEs. The volume is highly self-contained. No familiarity is assumed with differentiable manifolds and Riemannian geometry, making the book accessible to a broad audience of readers, including graduate students and researchers.

Some Nonlinear Problems in Riemannian Geometry
Some Nonlinear Problems in Riemannian Geometry

Author: Thierry Aubin

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 414

ISBN-13: 3662130068

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This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

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.

Convex Analysis and Nonlinear Geometric Elliptic Equations
Convex Analysis and Nonlinear Geometric Elliptic Equations

Author: Ilya J. Bakelman

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 524

ISBN-13: 3642698816

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Investigations in modem nonlinear analysis rely on ideas, methods and prob lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, ex emplary problems in nonlinear analysis were subject to intensive study and examination. The united ideas and methods of differential geometry, topology, differential equations and functional analysis as well as other areas of research in mathematics were successfully applied towards the complete solution of com plex problems in nonlinear analysis. It is not possible to encompass in the scope of one book all concepts, ideas, methods and results related to nonlinear analysis. Therefore, we shall restrict ourselves in this monograph to nonlinear elliptic boundary value problems as well as global geometric problems. In order that we may examine these prob lems, we are provided with a fundamental vehicle: The theory of convex bodies and hypersurfaces. In this book we systematically present a series of centrally significant results obtained in the second half of the twentieth century up to the present time. Particular attention is given to profound interconnections between various divisions in nonlinear analysis. The theory of convex functions and bodies plays a crucial role because the ellipticity of differential equations is closely connected with the local and global convexity properties of their solutions. Therefore it is necessary to have a sufficiently large amount of material devoted to the theory of convex bodies and functions and their connections with partial differential equations.