Download or read book The Monge Amp re Equation written by Cristian E. Gutiérrez and published by Birkhäuser. This book was released on 2016-10-22 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Monge Amp re Equation written by Cristian E. Gutierrez and published by Springer Science & Business Media. This book was released on 2001-05-11 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Monge-Ampère equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampère type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis – covering lemmas and set decompositions.
Download or read book Analysis of Monge Amp re Equations written by Nam Q. Le and published by American Mathematical Society. This book was released on 2024-03-08 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonlinear Analysis on Manifolds Monge Amp re Equations written by Thierry Aubin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Dynamical and Geometric Aspects of Hamilton Jacobi and Linearized Monge Amp re Equations written by Hiroyoshi Mitake and published by Springer. This book was released on 2017-06-14 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Consisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge–Ampère and linearized Monge–Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge–Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry. Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton–Jacobi equations, which have received much attention in the last two decades, and a new approach to the subject, the nonlinear adjoint method, is introduced. The appendix offers a short introduction to the theory of viscosity solutions of first-order Hamilton–Jacobi equations.
Download or read book Nonlinear Elasticity written by James Johnston Stoker and published by . This book was released on 1968 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonlinear PDEs Their Geometry and Applications written by Radosław A. Kycia and published by Springer. This book was released on 2019-05-18 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.
Download or read book The Complex Monge Ampere Equation and Pluripotential Theory written by Sławomir Kołodziej and published by American Mathematical Soc.. This book was released on 2005 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Multidimensional Residue Theory and Applications written by Alekos Vidras and published by American Mathematical Society. This book was released on 2023-10-18 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
Download or read book Differential Equations Geometry Symmetries and Integrability written by Boris Kruglikov and published by Springer Science & Business Media. This book was released on 2009-07-24 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.
Download or read book Applied Partial Differential Equations written by J. R. Ockendon and published by . This book was released on 2003 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: Partial differential equations are used in mathematical models of a huge range of real-world phenomena, from electromagnetism to financial markets. This new edition of Applied PDEs contains many new sections and exercises Including, American options, transform methods, free surface flows, linear elasticity and complex characteristics.
Download or read book Contact Geometry and Nonlinear Differential Equations written by Alexei Kushner and published by Cambridge University Press. This book was released on 2007 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.
Download or read book Advances in Analysis and Geometry written by Tao Qian and published by Birkhäuser. This book was released on 2012-12-06 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: At the heart of Clifford analysis is the study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool. This book focuses on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. This book collects refereed papers from a satellite conference to the ICM 2002, plus invited contributions. All articles contain unpublished new results.
Download or read book Regularity Theory for Quasilinear Elliptic Systems and Monge Ampere Equations in Two Dimensions written by Friedmar Schulz and published by Springer. This book was released on 2006-12-08 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes have been written as an introduction to the characteristic theory for two-dimensional Monge-Ampère equations, a theory largely developed by H. Lewy and E. Heinz which has never been presented in book form. An exposition of the Heinz-Lewy theory requires auxiliary material which can be found in various monographs, but which is presented here, in part because the focus is different, and also because these notes have an introductory character. Self-contained introductions to the regularity theory of elliptic systems, the theory of pseudoanalytic functions and the theory of conformal mappings are included. These notes grew out of a seminar given at the University of Kentucky in the fall of 1988 and are intended for graduate students and researchers interested in this area.
Download or read book Convex Analysis and Nonlinear Geometric Elliptic Equations written by Ilya J. Bakelman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Handbook of Geometric Analysis written by Lizhen Ji and published by . This book was released on 2008 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Geometric Analysis combines differential equations with differential geometry. An important aspect of geometric analysis is to approach geometric problems by studying differential equations. Besides some known linear differential operators such as the Laplace operator, many differential equations arising from differential geometry are nonlinear. A particularly important example is the Monge-Amperè equation. Applications to geometric problems have also motivated new methods and techniques in differential equations. The field of geometric analysis is broad and has had many striking applications. This handbook of geometric analysis--the first of the two to be published in the ALM series--presents introductions and survey papers treating important topics in geometric analysis, with their applications to related fields. It can be used as a reference by graduate students and by researchers in related areas."--Back cover.
Download or read book Monge Ampere Equation Applications to Geometry and Optimization written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1999 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
