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Young Measures And Compactness In Measure Spaces
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Book Synopsis Young Measures and Compactness in Measure Spaces by : Liviu C. Florescu
Download or read book Young Measures and Compactness in Measure Spaces written by Liviu C. Florescu and published by Walter de Gruyter. This book was released on 2012-05-29 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years, technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such optimal solution can not be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain such a relaxation by means of Young measures. This monograph is a self-contained book which gathers all theoretical aspects related to the defining of Young measures (measurability, disintegration, stable convergence, compactness), a book which is also a useful tool for those interested in theoretical foundations of the measure theory. It provides a complete set of classical and recent compactness results in measure and function spaces. The book is organized in three chapters: The first chapter covers background material on measure theory in abstract frame. In the second chapter the measure theory on topological spaces is presented. Compactness results from the first two chapters are used to study Young measures in the third chapter. All results are accompanied by full demonstrations and for many of these results different proofs are given. All statements are fully justified and proved.
Book Synopsis Young Measures on Topological Spaces by : Charles Castaing
Download or read book Young Measures on Topological Spaces written by Charles Castaing and published by Springer Science & Business Media. This book was released on 2006-04-11 with total page 327 pages. Available in PDF, EPUB and Kindle. Book excerpt: Young measures are now a widely used tool in the Calculus of Variations, in Control Theory, in Probability Theory and other fields. They are known under different names such as "relaxed controls," "fuzzy random variables" and many other names. This monograph provides a unified presentation of the theory, along with new results and applications in various fields. It can serve as a reference on the subject. Young measures are presented in a general setting which includes finite and for the first time infinite dimensional spaces: the fields of applications of Young measures (Control Theory, Calculus of Variations, Probability Theory...) are often concerned with problems in infinite dimensional settings. The theory of Young measures is now well understood in a finite dimensional setting, but open problems remain in the infinite dimensional case. We provide several new results in the general frame, which are new even in the finite dimensional setting, such as characterizations of convergence in measure of Young measures (Chapter 3) and compactness criteria (Chapter 4).These results are established under a different form (and with less details and developments) in recent papers by the same authors. We also provide new applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).
Book Synopsis Measure Theory by : Vladimir I. Bogachev
Download or read book Measure Theory written by Vladimir I. Bogachev and published by Springer Science & Business Media. This book was released on 2007-01-15 with total page 1075 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.
Book Synopsis Methods of Nonconvex Analysis by : Arrigo Cellina
Download or read book Methods of Nonconvex Analysis written by Arrigo Cellina and published by Springer. This book was released on 2006-11-14 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Compressible Navier-Stokes Equations by : Pavel Plotnikov
Download or read book Compressible Navier-Stokes Equations written by Pavel Plotnikov and published by Springer Science & Business Media. This book was released on 2012-08-04 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on the applications to aerodynamics. The topics covered include: modeling of compressible viscous flows; modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations; applications to optimal shape design in aerodynamics; kinetic theory for equations with oscillating data; new approach to the boundary value problems for transport equations. The monograph offers a comprehensive and self-contained introduction to recent mathematical tools designed to handle the problems arising in the theory.
Book Synopsis Weak Convergence of Measures by : Vladimir I. Bogachev
Download or read book Weak Convergence of Measures written by Vladimir I. Bogachev and published by American Mathematical Society. This book was released on 2024-07-29 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures. The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included. The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.
Book Synopsis Stochastically Forced Compressible Fluid Flows by : Dominic Breit
Download or read book Stochastically Forced Compressible Fluid Flows written by Dominic Breit and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-01-22 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains a first systematic study of compressible fluid flows subject to stochastic forcing. The bulk is the existence of dissipative martingale solutions to the stochastic compressible Navier-Stokes equations. These solutions are weak in the probabilistic sense as well as in the analytical sense. Moreover, the evolution of the energy can be controlled in terms of the initial energy. We analyze the behavior of solutions in short-time (where unique smooth solutions exists) as well as in the long term (existence of stationary solutions). Finally, we investigate the asymptotics with respect to several parameters of the model based on the energy inequality. Contents Part I: Preliminary results Elements of functional analysis Elements of stochastic analysis Part II: Existence theory Modeling fluid motion subject to random effects Global existence Local well-posedness Relative energy inequality and weak–strong uniqueness Part III: Applications Stationary solutions Singular limits
Book Synopsis Handbook of Multivalued Analysis by : Shouchuan Hu
Download or read book Handbook of Multivalued Analysis written by Shouchuan Hu and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 941 pages. Available in PDF, EPUB and Kindle. Book excerpt: In volume I we developed the tools of "Multivalued Analysis. " In this volume we examine the applications. After all, the initial impetus for the development of the theory of set-valued functions came from its applications in areas such as control theory and mathematical economics. In fact, the needs of control theory, in particular the study of systems with a priori feedback, led to the systematic investigation of differential equations with a multi valued vector field (differential inclusions). For this reason, we start this volume with three chapters devoted to set-valued differential equations. However, in contrast to the existing books on the subject (i. e. J. -P. Aubin - A. Cellina: "Differential Inclusions," Springer-Verlag, 1983, and Deimling: "Multivalued Differential Equations," W. De Gruyter, 1992), here we focus on "Evolution Inclusions," which are evolution equations with multi valued terms. Evolution equations were raised to prominence with the development of the linear semigroup theory by Hille and Yosida initially, with subsequent im portant contributions by Kato, Phillips and Lions. This theory allowed a successful unified treatment of some apparently different classes of nonstationary linear par tial differential equations and linear functional equations. The needs of dealing with applied problems and the natural tendency to extend the linear theory to the nonlinear case led to the development of the nonlinear semigroup theory, which became a very effective tool in the analysis of broad classes of nonlinear evolution equations.
Book Synopsis Proceedings of the International Congress of Mathematicians by : S.D. Chatterji
Download or read book Proceedings of the International Congress of Mathematicians written by S.D. Chatterji and published by Birkhäuser. This book was released on 2012-12-06 with total page 1669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)
Book Synopsis Measure Theory and Nonlinear Evolution Equations by : Flavia Smarrazzo
Download or read book Measure Theory and Nonlinear Evolution Equations written by Flavia Smarrazzo and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-04-19 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on measure theory with applications to partial differential equations covers general measure theory, Lebesgue spaces of real-valued and vector-valued functions, different notions of measurability for the latter, weak convergence of functions and measures, Radon and Young measures, capacity. A comprehensive discussion of applications to quasilinear parabolic and hyperbolic problems is provided.
Book Synopsis Advances in Mathematical Economics by : Shigeo Kusuoka
Download or read book Advances in Mathematical Economics written by Shigeo Kusuoka and published by Springer. This book was released on 2018-07-31 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: The series is designed to bring together those mathematicians who are seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking effective mathematical tools for their research. A lot of economic problems can be formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories.
Book Synopsis Geometric Control and Nonsmooth Analysis by : Fabio Ancona
Download or read book Geometric Control and Nonsmooth Analysis written by Fabio Ancona and published by World Scientific. This book was released on 2008 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this volume is to provide a synthetic account of past research, to give an up-to-date guide to current intertwined developments of control theory and nonsmooth analysis, and also to point to future research directions.
Book Synopsis Nonlinear Analysis - Theory and Methods by : Nikolaos S. Papageorgiou
Download or read book Nonlinear Analysis - Theory and Methods written by Nikolaos S. Papageorgiou and published by Springer. This book was released on 2019-02-26 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.
Download or read book Gradient Flows written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2008-10-29 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.
Book Synopsis Variational Analysis in Sobolev and BV Spaces by : Hedy Attouch
Download or read book Variational Analysis in Sobolev and BV Spaces written by Hedy Attouch and published by SIAM. This book was released on 2014-10-02 with total page 794 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is an excellent guide for anyone interested in variational analysis, optimization, and PDEs. It offers a detailed presentation of the most important tools in variational analysis as well as applications to problems in geometry, mechanics, elasticity, and computer vision. This second edition covers several new topics: new section on capacity theory and elements of potential theory now includes the concepts of quasi-open sets and quasi-continuity; increased number of examples in the areas of linearized elasticity system, obstacles problems, convection-diffusion, and semilinear equations; new section on mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; new subsection on stochastic homogenization establishes the mathematical tools coming from ergodic theory; and an entirely new and comprehensive chapter (17) devoted to gradient flows and the dynamical approach to equilibria. The book is intended for Ph.D. students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way.
Book Synopsis Differential Equations, Chaos and Variational Problems by : Vasile Staicu
Download or read book Differential Equations, Chaos and Variational Problems written by Vasile Staicu and published by Springer Science & Business Media. This book was released on 2008-03-12 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of original articles and surveys written by leading experts in their fields is dedicated to Arrigo Cellina and James A. Yorke on the occasion of their 65th birthday. The volume brings the reader to the border of research in differential equations, a fast evolving branch of mathematics that, besides being a main subject for mathematicians, is one of the mathematical tools most used both by scientists and engineers.
Book Synopsis Advances in Mathematical Economics by : Toru Maruyama
Download or read book Advances in Mathematical Economics written by Toru Maruyama and published by Springer Nature. This book was released on 2020-02-20 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: The series is designed to bring together those mathematicians who are seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking effective mathematical tools for their research. A lot of economic problems can be formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories.
