Progress in Nonlinear Differential Equations and Their Applications

Extensions of Moser–Bangert Theory

Locally Minimal Solutions

Authors: Rabinowitz, Paul H., Stredulinsky, Edward W.

  • Outgrowth of Moser–Bangert's work on solutions of a family of nonlinear elliptic partial differential equations
  • Develops and examines the rich structure of the set of solutions of the simpler model case (PDE)
  • Minimization arguments are an important tool in the investigation
  • Unique book in the literature
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Hardcover $169.00
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About this book

With the goal of establishing a version for partial differential equations (PDEs) of the Aubry–Mather theory of monotone twist maps, Moser and then Bangert studied solutions of their model equations that possessed certain minimality and monotonicity properties. This monograph presents extensions of the Moser–Bangert approach that include solutions of a family of nonlinear elliptic PDEs on Rn and an Allen–Cahn PDE model of phase transitions.

After recalling the relevant Moser–Bangert results, Extensions of Moser–Bangert Theory pursues the rich structure of the set of solutions of a simpler model case, expanding upon the studies of Moser and Bangert to include solutions that merely have local minimality properties. Subsequent chapters build upon the introductory results, making the monograph self contained.

Part I introduces a variational approach involving a renormalized functional to characterize the basic heteroclinic solutions obtained by Bangert. Following that, Parts II and III employ these basic solutions together with constrained minimization methods to construct multitransition heteroclinic and homoclinic solutions on R×Tn-1 and R2×Tn-2, respectively, as local minima of the renormalized functional. The work is intended for mathematicians who specialize in partial differential equations and may also be used as a text for a graduate topics course in PDEs.

Reviews

From the reviews:

“This book contains a study of the solution set to (PDE), expanding work by Moser and Bangert and previous work by the authors for (AC). … This is an important piece of work concerning a difficult and deep matter. … This a very good demonstration of the power of variational methods, showing that they can be modified, extended and combined in order to deal with many different kinds of problems.” (Jesús Hernández, Mathematical Reviews, Issue 2012 m)


Table of contents (13 chapters)

  • Introduction

    Rabinowitz, Paul H. (et al.)

    Pages 1-6

  • Function Spaces and the First Renormalized Functional

    Rabinowitz, Paul H. (et al.)

    Pages 9-22

  • The Simplest Heteroclinics

    Rabinowitz, Paul H. (et al.)

    Pages 23-35

  • Heteroclinics in x 1 and x 2

    Rabinowitz, Paul H. (et al.)

    Pages 37-52

  • More Basic Solutions

    Rabinowitz, Paul H. (et al.)

    Pages 53-62

Buy this book

eBook $129.00
price for USA (gross)
  • ISBN 978-0-8176-8117-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $169.00
price for USA
  • ISBN 978-0-8176-8116-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Extensions of Moser–Bangert Theory
Book Subtitle
Locally Minimal Solutions
Authors
Series Title
Progress in Nonlinear Differential Equations and Their Applications
Series Volume
81
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-0-8176-8117-3
DOI
10.1007/978-0-8176-8117-3
Hardcover ISBN
978-0-8176-8116-6
Series ISSN
1421-1750
Edition Number
1
Number of Pages
VIII, 208
Topics