Modern Methods In The Calculus Of Variations

Author : Irene Fonseca,Giovanni Leoni
Publisher : Springer Science & Business Media
Page : 602 pages
File Size : 52,8 Mb
Release : 2007-08-22
Category : Science
ISBN : 9780387690063

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Modern Methods in the Calculus of Variations by Irene Fonseca,Giovanni Leoni Pdf

This is the first of two books on methods and techniques in the calculus of variations. Contemporary arguments are used throughout the text to streamline and present in a unified way classical results, and to provide novel contributions at the forefront of the theory. This book addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in L^p spaces. It prepares the ground for the second volume where the variational treatment of functionals involving fields and their derivatives will be undertaken within the framework of Sobolev spaces. This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. Irene Fonseca is the Mellon College of Science Professor of Mathematics and is currently the Director of the Center for Nonlinear Analysis in the Department of Mathematical Sciences at Carnegie Mellon University. Her research interests lie in the areas of continuum mechanics, calculus of variations, geometric measure theory and partial differential equations. Giovanni Leoni is also a professor in the Department of Mathematical Sciences at Carnegie Mellon University. He focuses his research on calculus of variations, partial differential equations and geometric measure theory with special emphasis on applications to problems in continuum mechanics and in materials science.

Author : Irene Fonseca
Publisher : Unknown
Page : 0 pages
File Size : 48,8 Mb
Release : 2007
Category : Electronic
ISBN : OCLC:1152557172

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Modern Methods in the Calculus of Variations: Lp Spaces by Irene Fonseca Pdf

Author : I. M. Gelfand,S. V. Fomin
Publisher : Courier Corporation
Page : 240 pages
File Size : 49,5 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486135014

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Calculus of Variations by I. M. Gelfand,S. V. Fomin Pdf

Fresh, lively text serves as a modern introduction to the subject, with applications to the mechanics of systems with a finite number of degrees of freedom. Ideal for math and physics students.

Author : Enrico Giusti
Publisher : World Scientific
Page : 412 pages
File Size : 47,8 Mb
Release : 2003
Category : Mathematics
ISBN : 9789812380432

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Direct Methods in the Calculus of Variations by Enrico Giusti Pdf

This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.

Author : Bernard Dacorogna
Publisher : Springer Science & Business Media
Page : 616 pages
File Size : 53,7 Mb
Release : 2007-11-21
Category : Mathematics
ISBN : 9780387552491

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Direct Methods in the Calculus of Variations by Bernard Dacorogna Pdf

This book is developed for the study of vectorial problems in the calculus of variations. The subject is a very active one and almost half of the book consists of new material. This is a new edition of the earlier book published in 1989 and it is suitable for graduate students. The book has been updated with some new material and examples added. Applications are included.

Author : Hans Sagan
Publisher : Courier Corporation
Page : 484 pages
File Size : 46,5 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486138022

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Introduction to the Calculus of Variations by Hans Sagan Pdf

Provides a thorough understanding of calculus of variations and prepares readers for the study of modern optimal control theory. Selected variational problems and over 400 exercises. Bibliography. 1969 edition.

Author : Robert Weinstock
Publisher : Courier Corporation
Page : 354 pages
File Size : 40,7 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486141060

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Calculus of Variations by Robert Weinstock Pdf

This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians. The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations. Later chapters cover isoperimetric problems, geometrical optics, Fermat's principle, dynamics of particles, the Sturm-Liouville eigenvalue-eigenfunction problem, the theory of elasticity, quantum mechanics, and electrostatics. Each chapter ends with a series of exercises which should prove very useful in determining whether the material in that chapter has been thoroughly grasped. The clarity of exposition makes this book easily accessible to anyone who has mastered first-year calculus with some exposure to ordinary differential equations. Physicists and engineers who find variational methods evasive at times will find this book particularly helpful. "I regard this as a very useful book which I shall refer to frequently in the future." J. L. Synge, Bulletin of the American Mathematical Society.

Author : Hansjörg Kielhöfer
Publisher : Springer
Page : 227 pages
File Size : 44,9 Mb
Release : 2018-01-25
Category : Mathematics
ISBN : 9783319711232

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Calculus of Variations by Hansjörg Kielhöfer Pdf

This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself. Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study. The book will be particularly useful for beginning graduate students from the physical, engineering, and mathematical sciences with a rigorous theoretical background.

Author : Filip Rindler
Publisher : Springer
Page : 444 pages
File Size : 52,5 Mb
Release : 2018-06-20
Category : Mathematics
ISBN : 9783319776378

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Calculus of Variations by Filip Rindler Pdf

This textbook provides a comprehensive introduction to the classical and modern calculus of variations, serving as a useful reference to advanced undergraduate and graduate students as well as researchers in the field. Starting from ten motivational examples, the book begins with the most important aspects of the classical theory, including the Direct Method, the Euler-Lagrange equation, Lagrange multipliers, Noether’s Theorem and some regularity theory. Based on the efficient Young measure approach, the author then discusses the vectorial theory of integral functionals, including quasiconvexity, polyconvexity, and relaxation. In the second part, more recent material such as rigidity in differential inclusions, microstructure, convex integration, singularities in measures, functionals defined on functions of bounded variation (BV), and Γ-convergence for phase transitions and homogenization are explored. While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study. The reader is assumed to be familiar with basic vector analysis, functional analysis, Sobolev spaces, and measure theory, though most of the preliminaries are also recalled in the appendix.

Author : Michael Struwe
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 54,7 Mb
Release : 2013-04-17
Category : Science
ISBN : 9783662032121

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Variational Methods by Michael Struwe Pdf

Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radò. The book gives a concise introduction to variational methods and presents an overview of areas of current research in this field. This new edition has been substantially enlarged, a new chapter on the Yamabe problem has been added and the references have been updated. All topics are illustrated by carefully chosen examples, representing the current state of the art in their field.

Author : Paolo Freguglia,Mariano Giaquinta
Publisher : Birkhäuser
Page : 293 pages
File Size : 46,6 Mb
Release : 2016-06-27
Category : Mathematics
ISBN : 9783319389455

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The Early Period of the Calculus of Variations by Paolo Freguglia,Mariano Giaquinta Pdf

This monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli’s work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange’s δ-calculus, including: • The isoperimetrical problems • Shortest lines and geodesics • Euler’s Methodus Inveniendi and the two Additamenta Finally, the authors give the readers a sense of how vast the calculus of variations has become in centuries hence, providing some idea of what lies outside the scope of the book as well as the current state of affairs in the field. This book will be of interest to anyone studying the calculus of variations who wants a deeper intuition for the techniques and ideas that are used, as well as historians of science and mathematics interested in the development and evolution of modern calculus and analysis.

Author : Agnieszka B Malinowska,Delfim F M Torres
Publisher : World Scientific Publishing Company
Page : 292 pages
File Size : 40,8 Mb
Release : 2012-09-14
Category : Mathematics
ISBN : 9781848169685

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Introduction to the Fractional Calculus of Variations by Agnieszka B Malinowska,Delfim F M Torres Pdf

This invaluable book provides a broad introduction to the fascinating and beautiful subject of Fractional Calculus of Variations (FCV). In 1996, FVC evolved in order to better describe non-conservative systems in mechanics. The inclusion of non-conservatism is extremely important from the point of view of applications. Forces that do not store energy are always present in real systems. They remove energy from the systems and, as a consequence, Noether's conservation laws cease to be valid. However, it is still possible to obtain the validity of Noether's principle using FCV. The new theory provides a more realistic approach to physics, allowing us to consider non-conservative systems in a natural way. The authors prove the necessary Euler–Lagrange conditions and corresponding Noether theorems for several types of fractional variational problems, with and without constraints, using Lagrangian and Hamiltonian formalisms. Sufficient optimality conditions are also obtained under convexity, and Leitmann's direct method is discussed within the framework of FCV. The book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and ambitious undergraduates in mathematics and mechanics. It provides an opportunity for an introduction to FCV for experienced researchers. The explanations in the book are detailed, in order to capture the interest of the curious reader, and the book provides the necessary background material required to go further into the subject and explore the rich research literature.

Author : Jürgen Jost,Xianqing Li-Jost
Publisher : Cambridge University Press
Page : 348 pages
File Size : 40,5 Mb
Release : 1998
Category : Mathematics
ISBN : 0521642035

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Calculus of Variations by Jürgen Jost,Xianqing Li-Jost Pdf

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Author : Bernard Dacorogna
Publisher : World Scientific Publishing Company
Page : 324 pages
File Size : 46,5 Mb
Release : 2014-08-13
Category : Mathematics
ISBN : 9781783265541

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Introduction to the Calculus of Variations by Bernard Dacorogna Pdf

The calculus of variations is one of the oldest subjects in mathematics, and it is very much alive and still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology. This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist — mathematicians, physicists, engineers, students or researchers — in discovering the subject's most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions. In this new edition, several new exercises have been added. The book, containing a total of 119 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.

Author : Michael Grinfeld
Publisher : John Wiley & Sons
Page : 634 pages
File Size : 43,5 Mb
Release : 2015-01-12
Category : Science
ISBN : 9783527411887

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Mathematical Tools for Physicists by Michael Grinfeld Pdf

The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.