Methods Of Geometry In The Theory Of Partial Differential Equations Principle Of The Cancellation Of Singularities

Author : Takashi Suzuki
Publisher : World Scientific
Page : 414 pages
File Size : 45,5 Mb
Release : 2024-01-22
Category : Mathematics
ISBN : 9789811287916

Get Book

Methods Of Geometry In The Theory Of Partial Differential Equations: Principle Of The Cancellation Of Singularities by Takashi Suzuki Pdf

Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena.In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently.This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis.Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d - δ systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering.The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas.

Author : Demetrios Christodoulou
Publisher : Princeton University Press
Page : 328 pages
File Size : 48,8 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882687

Get Book

The Action Principle and Partial Differential Equations. (AM-146), Volume 146 by Demetrios Christodoulou Pdf

This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Author : Agostino Prastaro,Themistocles M. Rassias
Publisher : World Scientific
Page : 482 pages
File Size : 45,9 Mb
Release : 1994
Category : Mathematics
ISBN : 9810214073

Get Book

Geometry in Partial Differential Equations by Agostino Prastaro,Themistocles M. Rassias Pdf

This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.

Author : Alexander Komech,Andrew Komech
Publisher : Springer Science & Business Media
Page : 165 pages
File Size : 55,7 Mb
Release : 2009-10-05
Category : Mathematics
ISBN : 9781441910950

Get Book

Principles of Partial Differential Equations by Alexander Komech,Andrew Komech Pdf

This concise book covers the classical tools of Partial Differential Equations Theory in today’s science and engineering. The rigorous theoretical presentation includes many hints, and the book contains many illustrative applications from physics.

Author : Alessio Figalli,Enrico Valdinoci,Ireneo Peral
Publisher : Springer
Page : 216 pages
File Size : 52,8 Mb
Release : 2018-05-23
Category : Mathematics
ISBN : 9783319740423

Get Book

Partial Differential Equations and Geometric Measure Theory by Alessio Figalli,Enrico Valdinoci,Ireneo Peral Pdf

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2–7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.

Author : Jean Pierre Bourguignon,Paolo de Bartolomeis,Mariano Giaquinta
Publisher : Cambridge University Press
Page : 198 pages
File Size : 46,7 Mb
Release : 1998-05-28
Category : Mathematics
ISBN : 0521632463

Get Book

Geometric Theory of Singular Phenomena in Partial Differential Equations by Jean Pierre Bourguignon,Paolo de Bartolomeis,Mariano Giaquinta Pdf

This book gathers together papers from a workshop held in Cortona, Italy. The contributions come from a group of outstanding mathematicians and together they cover the most recent advances in the geometric theory of singular phenomena of partial differential equations occurring in real and complex differential geometry. This volume will be of great interest to all those whose research interests lie in real and complex differential geometry, partial differential equations, and gauge theory.

Author : Walter A. Strauss
Publisher : John Wiley & Sons
Page : 467 pages
File Size : 44,9 Mb
Release : 2007-12-21
Category : Mathematics
ISBN : 9780470054567

Get Book

Partial Differential Equations by Walter A. Strauss Pdf

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

Author : Robert Hardt
Publisher : American Mathematical Soc.
Page : 356 pages
File Size : 47,6 Mb
Release : 1996
Category : Mathematics
ISBN : 0821804316

Get Book

Nonlinear partial differential equations in differential geometry by Robert Hardt Pdf

This book contains lecture notes of minicourses at the Regional Geometry Institute at Park City, Utah, in July 1992. Presented here are surveys of breaking developments in a number of areas of nonlinear partial differential equations in differential geometry. The authors of the articles are not only excellent expositors, but are also leaders in this field of research. All of the articles provide in-depth treatment of the topics and require few prerequisites and less background than current research articles.

Author : V V Zharinov
Publisher : World Scientific
Page : 372 pages
File Size : 40,6 Mb
Release : 1992-03-26
Category : Mathematics
ISBN : 9789814513999

Get Book

Lecture Notes on Geometrical Aspects of Partial Differential Equations by V V Zharinov Pdf

This book focuses on the properties of nonlinear systems of PDE with geometrical origin and the natural description in the language of infinite-dimensional differential geometry. The treatment is very informal and the theory is illustrated by various examples from mathematical physics. All necessary information about the infinite-dimensional geometry is given in the text. Contents:Introduction: Internal Geometry of PDE:Differential ManifoldsLie-Backlund MappingsLie-Backlund Fields and Infinitesimal SymmetriesCartan Forms, Currents and Conservation LawsC-Spectral Sequence. Further Properties of Conservation LawsTrivial Equations. The Formal Variational CalculusEvolution EquationsExternal Geometry of PDE:Differential SubmanifoldsNormal Projection. External Fields and FormsTrivial Ambient Differential ManifoldsThe Characteristic MappingThe Green's FormulaLow-Dimensional Conservation LawsBacklund CorrespondenceFurther Studies:Lagrangian FormalismHamiltonian EquationsExample: The Nambu's StringAppendix Readership: Graduate students and researchers in mathematical physics. keywords:Differential Manifolds;Lie-Bäcklund Mappings;Cartan Forms;Currents;Conservation Laws;Lagrangian Formation;Hamiltonian Equations

Author : S.S. Chern
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 47,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211105

Get Book

Seminar on Nonlinear Partial Differential Equations by S.S. Chern Pdf

When the Mathematical Sciences Research Institute was started in the Fall of 1982, one of the programs was "non-linear partial differential equations". A seminar was organized whose audience consisted of graduate students of the University and mature mathematicians who are not experts in the field. This volume contains 18 of these lectures. An effort is made to have an adequate Bibliography for further information. The Editor wishes to take this opportunity to thank all the speakers and the authors of the articles presented in this volume for their cooperation. S. S. Chern, Editor Table of Contents Geometrical and Analytical Questions Stuart S. Antman 1 in Nonlinear Elasticity An Introduction to Euler's Equations Alexandre J. Chorin 31 for an Incompressible Fluid Linearizing Flows and a Cohomology Phillip Griffiths 37 Interpretation of Lax Equations The Ricci Curvature Equation Richard Hamilton 47 A Walk Through Partial Differential Fritz John 73 Equations Remarks on Zero Viscosity Limit for Tosio Kato 85 Nonstationary Navier-Stokes Flows with Boundary Free Boundary Problems in Mechanics Joseph B. Keller 99 The Method of Partial Regularity as Robert V.

Author : Joseph Grifone,Zoltán Muzsnay
Publisher : World Scientific
Page : 228 pages
File Size : 42,8 Mb
Release : 2000-05-25
Category : Mathematics
ISBN : 9789814495363

Get Book

Variational Principles for Second-Order Differential Equations by Joseph Grifone,Zoltán Muzsnay Pdf

The inverse problem of the calculus of variations was first studied by Helmholtz in 1887 and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations. This book looks at second-order differential equations and asks if they can be written as Euler–Lagrangian equations. If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi–Civita for some Riemann metric. To solve the inverse problem, the authors use the formal integrability theory of overdetermined partial differential systems in the Spencer–Quillen–Goldschmidt version. The main theorems of the book furnish a complete illustration of these techniques because all possible situations appear: involutivity, 2-acyclicity, prolongation, computation of Spencer cohomology, computation of the torsion, etc. Contents:An Introduction to Formal Integrability Theory of Partial Differential SystemsFrölicher–Nijenhuis Theory of DerivationsDifferential Algebraic Formalism of ConnectionsNecessary Conditions for Variational SpraysObstructions to the Integrability of the Euler–Lagrange SystemThe Classification of Locally Variational Sprays on Two-Dimensional ManifoldsEuler–Lagrange Systems in the Isotropic Case Readership: Mathematicians. Keywords:Calculus of Variations;Inverse Problem;Euler-Lagrange Equation;Sprays;Formal Integrability;Involution;Janet-Riquier Theory;Spencer TheoryReviews: “Everybody seriously interested in the modern theory of the inverse problem of the calculus of variations should take a look at this book.” Zentralblatt MATH

Author : Christopher I. Byrnes
Publisher : Marcel Dekker
Page : 348 pages
File Size : 50,5 Mb
Release : 1979
Category : Mathematics
ISBN : UOM:39015049311767

Get Book

Partial Differential Equations and Geometry by Christopher I. Byrnes Pdf

Author : Anonim
Publisher : Unknown
Page : 348 pages
File Size : 45,5 Mb
Release : 1992
Category : Mechanics, Applied
ISBN : OSU:32435026160556

Get Book

Applied Mechanics Reviews by Anonim Pdf

Author : Stefan Hildebrandt,Hermann Karcher
Publisher : Springer Science & Business Media
Page : 663 pages
File Size : 47,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642556272

Get Book

Geometric Analysis and Nonlinear Partial Differential Equations by Stefan Hildebrandt,Hermann Karcher Pdf

This book is not a textbook, but rather a coherent collection of papers from the field of partial differential equations. Nevertheless we believe that it may very well serve as a good introduction into some topics of this classical field of analysis which, despite of its long history, is highly modem and well prospering. Richard Courant wrote in 1950: "It has always been a temptationfor mathematicians to present the crystallized product of their thought as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods or more general significance. " We think that many, if not all, papers of this book are written in this spirit and will give the reader access to an important branch of analysis by exhibiting interest ing problems worth to be studied. Most of the collected articles have an extensive introductory part describing the history of the presented problems as well as the state of the art and offer a well chosen guide to the literature. This way the papers became lengthier than customary these days, but the level of presentation is such that an advanced graduate student should find the various articles both readable and stimulating.

Author : R.P. Gilbert
Publisher : Springer
Page : 405 pages
File Size : 43,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540379539

Get Book

Constructive Methods for Elliptic Equations by R.P. Gilbert Pdf

Primarily these lectures are a report on recent work by the Indiana University group working on Function theoretic methods as applied to the theory of partial differential equations.