Calculus Of Variations And Harmonic Maps

Author : Hajime Urakawa
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 45,8 Mb
Release : 2013-02-15
Category : Mathematics
ISBN : 9780821894132

Get Book

Calculus of Variations and Harmonic Maps by Hajime Urakawa Pdf

This book provides a wide view of the calculus of variations as it plays an essential role in various areas of mathematics and science. Containing many examples, open problems, and exercises with complete solutions, the book would be suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods. The first part of the book is devoted to explaining the notion of (infinite-dimensional) manifolds and contains many examples. An introduction to Morse theory of Banach manifolds is provided, along with a proof of the existence of minimizing functions under the Palais-Smale condition. The second part, which may be read independently of the first, presents the theory of harmonic maps, with a careful calculation of the first and second variations of the energy. Several applications of the second variation and classification theories of harmonic maps are given.

Author : Martin Fuchs
Publisher : Springer Science & Business Media
Page : 155 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9783322865281

Get Book

Topics in the Calculus of Variations by Martin Fuchs Pdf

This book illustrates two basic principles in the calculus of variations which are the question of existence of solutions and closely related the problem of regularity of minimizers. Chapter one studies variational problems for nonquadratic energy functionals defined on suitable classes of vectorvalued functions where also nonlinear constraints are incorporated. Problems of this type arise for mappings between Riemannian manifolds or in nonlinear elasticity. Using direct methods the existence of generalized minimizers is rather easy to establish and it is then shown that regularity holds up to a set of small measure. Chapter two contains a short introduction into Geometric Measure Theory which serves as a basis for developing an existence theory for (generalized) manifolds with prescribed mean curvature form and boundary in arbitrary dimensions and codimensions. One major aspect of the book is to concentrate on techniques and to present methods which turn out to be useful for applications in regularity theorems as well as for existence problems.

Author : Mariano Giaquinta,Giuseppe Modica,Jiri Soucek
Publisher : Springer Science & Business Media
Page : 744 pages
File Size : 51,8 Mb
Release : 1998-08-19
Category : Mathematics
ISBN : 3540640096

Get Book

Cartesian Currents in the Calculus of Variations I by Mariano Giaquinta,Giuseppe Modica,Jiri Soucek Pdf

This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph

Author : Eric Loubeau,Stefano Montaldo
Publisher : American Mathematical Soc.
Page : 296 pages
File Size : 49,5 Mb
Release : 2011
Category : Geometry, Differential
ISBN : 9780821849873

Get Book

Harmonic Maps and Differential Geometry by Eric Loubeau,Stefano Montaldo Pdf

This volume contains the proceedings of a conference held in Cagliari, Italy, from September 7-10, 2009, to celebrate John C. Wood's 60th birthday. These papers reflect the many facets of the theory of harmonic maps and its links and connections with other topics in Differential and Riemannian Geometry. Two long reports, one on constant mean curvature surfaces by F. Pedit and the other on the construction of harmonic maps by J. C. Wood, open the proceedings. These are followed by a mix of surveys on Prof. Wood's area of expertise: Lagrangian surfaces, biharmonic maps, locally conformally Kahler manifolds and the DDVV conjecture, as well as several research papers on harmonic maps. Other research papers in the volume are devoted to Willmore surfaces, Goldstein-Pedrich flows, contact pairs, prescribed Ricci curvature, conformal fibrations, the Fadeev-Hopf model, the Compact Support Principle and the curvature of surfaces.

Author : Seiki Nishikawa,Richard Schoen
Publisher : Springer Science & Business Media
Page : 160 pages
File Size : 40,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9784431684022

Get Book

Lectures on Geometric Variational Problems by Seiki Nishikawa,Richard Schoen Pdf

In this volume are collected notes of lectures delivered at the First In ternational Research Institute of the Mathematical Society of Japan. This conference, held at Tohoku University in July 1993, was devoted to geometry and global analysis. Subsequent to the conference, in answer to popular de mand from the participants, it was decided to publish the notes of the survey lectures. Written by the lecturers themselves, all experts in their respective fields, these notes are here presented in a single volume. It is hoped that they will provide a vivid account of the current research, from the introduc tory level up to and including the most recent results, and will indicate the direction to be taken by future researeh. This compilation begins with Jean-Pierre Bourguignon's notes entitled "An Introduction to Geometric Variational Problems," illustrating the gen eral framework of the field with many examples and providing the reader with a broad view of the current research. Following this, Kenji Fukaya's notes on "Geometry of Gauge Fields" are concerned with gauge theory and its applications to low-dimensional topology, without delving too deeply into technical detail. Special emphasis is placed on explaining the ideas of infi nite dimensional geometry that, in the literature, are often hidden behind rigorous formulations or technical arguments.

Author : Mariano Giaquinta,Guiseppe Modica,Jiri Soucek
Publisher : Springer Science & Business Media
Page : 728 pages
File Size : 40,5 Mb
Release : 1998-08-19
Category : Mathematics
ISBN : 354064010X

Get Book

Cartesian Currents in the Calculus of Variations II by Mariano Giaquinta,Guiseppe Modica,Jiri Soucek Pdf

This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph

Author : Seiki Nishikawa
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 47,7 Mb
Release : 2002
Category : Mathematics
ISBN : 0821813560

Get Book

Kikagakuteki Henbun Mondai by Seiki Nishikawa Pdf

A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus. Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge. This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps. The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an in-depth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation. The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of Eells-Sampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as self-contained as possible for easy reading. Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter. This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.

Author : James Eells,Luc Lemaire
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 43,7 Mb
Release : 1983-01-01
Category : Mathematics
ISBN : 0821888951

Get Book

Selected Topics in Harmonic Maps by James Eells,Luc Lemaire Pdf

Author : Stefan Hildebrandt,Rolf Leis
Publisher : Springer
Page : 433 pages
File Size : 55,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540460244

Get Book

Partial Differential Equations and Calculus of Variations by Stefan Hildebrandt,Rolf Leis Pdf

This volume contains 18 invited papers by members and guests of the former Sonderforschungsbereich in Bonn (SFB 72) who, over the years, collaborated on the research group "Solution of PDE's and Calculus of Variations". The emphasis is on existence and regularity results, on special equations of mathematical physics and on scattering theory.

Author : John C. Wood
Publisher : Springer-Verlag
Page : 328 pages
File Size : 55,5 Mb
Release : 2013-07-02
Category : Mathematics
ISBN : 9783663140924

Get Book

Harmonic Maps and Integrable Systems by John C. Wood Pdf

Author : Mariano Giaquinta
Publisher : Springer
Page : 194 pages
File Size : 48,9 Mb
Release : 2006-11-14
Category : Science
ISBN : 9783540460756

Get Book

Topics in Calculus of Variations by Mariano Giaquinta Pdf

Author : Mariano Giaquinta,Guiseppe Modica,Jiri Soucek
Publisher : Springer Science & Business Media
Page : 717 pages
File Size : 51,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662062180

Get Book

Cartesian Currents in the Calculus of Variations II by Mariano Giaquinta,Guiseppe Modica,Jiri Soucek Pdf

Non-scalar variational problems appear in different fields. In geometry, for in stance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of a-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i. e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial.

Author : Ju rgen Jost
Publisher : Unknown
Page : 426 pages
File Size : 48,8 Mb
Release : 1996
Category : Mathematics
ISBN : UOM:39015042035322

Get Book

Geometric Analysis and the Calculus of Variations by Ju rgen Jost Pdf

This volume is dedicated to the ideas of Stefan Hildebrant, whose doctrinal students include Bernd Schmidt and Klaus Stefan. His solution to the boundry regularity question for minimal surfaces bounded by a pescribed Jordan curve brought him world fame.

Author : Mariano Giaquinta
Publisher : Princeton University Press
Page : 312 pages
File Size : 47,6 Mb
Release : 1983-11-21
Category : Mathematics
ISBN : 0691083312

Get Book

Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems by Mariano Giaquinta Pdf

The description for this book, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105, will be forthcoming.

Author : Mariano Giaquinta,Luca Martinazzi
Publisher : Springer Science & Business Media
Page : 370 pages
File Size : 44,7 Mb
Release : 2013-07-30
Category : Mathematics
ISBN : 9788876424434

Get Book

An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by Mariano Giaquinta,Luca Martinazzi Pdf

This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered during the International Congress of Mathematicians in 1900 in Paris: 19th problem: Are the solutions to regular problems in the Calculus of Variations always necessarily analytic? 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and L^p-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; energy minimizing harmonic maps and minimal graphs in codimension 1 and greater than 1. In this second deeply revised edition we also included the regularity of 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.