Extensions of Moser–Bangert Theory

Locally Minimal Solutions

  • Paul H. Rabinowitz
  • Edward W. Stredulinsky

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 81)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Paul H. Rabinowitz, Edward W. Stredulinsky
    Pages 1-6
  3. Basic Solutions

    1. Front Matter
      Pages 7-7
    2. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 9-22
    3. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 23-35
    4. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 37-52
    5. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 53-62
  4. Shadowing Results

    1. Front Matter
      Pages 63-63
    2. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 65-79
    3. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 81-87
    4. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 89-96
    5. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 97-118
    6. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 119-129
    7. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 131-153
  5. Solutions of (PDE) Defined on $$\mathbb{R}^2 \times \mathbb{T}^{n-2}$$

    1. Front Matter
      Pages 155-155
    2. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 157-177
    3. Paul H. Rabinowitz, Edward W. Stredulinsky
      Pages 179-203
  6. Back Matter
    Pages 205-208

About this book

Introduction

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.

Keywords

Allen-Cahn PDE model Moser-Bangert solutions function spaces heteroclinic behavior monotone multitransition solutions renormalized functional shadowing results

Authors and affiliations

  • Paul H. Rabinowitz
    • 1
  • Edward W. Stredulinsky
    • 2
  1. 1., Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA
  2. 2., Department of MathematicsUniversity of Wisconsin–Rock CountyJanesvilleUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-8117-3
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-8116-6
  • Online ISBN 978-0-8176-8117-3
  • About this book