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Elliptic Systems of Phase Transition Type

  • Nicholas D. Alikakos
  • Giorgio Fusco
  • Panayotis Smyrnelis

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 91)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 1-23
  3. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 25-86
  4. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 87-109
  5. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 111-133
  6. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 135-180
  7. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 181-205
  8. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 207-240
  9. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 241-280
  10. Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
    Pages 281-336
  11. Back Matter
    Pages 337-343

About this book

Introduction

This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes – non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates.

Key features and topics of this self-contained, systematic exposition include:

• Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions.

• Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves.

• Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates.

• Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results.

This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations – ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science.

Keywords

geodesics standing waves maximum principle point group crystalline

Authors and affiliations

  • Nicholas D. Alikakos
    • 1
  • Giorgio Fusco
    • 2
  • Panayotis Smyrnelis
    • 3
  1. 1.Department of MathematicsNational and Kapodistrian UniversityAthensGreece
  2. 2.Department of MathematicsUniversity of L’AquilaCoppitoItaly
  3. 3.Center for Mathematical ModelingUniversity of ChileSantiagoChile

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-90572-3
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-90571-6
  • Online ISBN 978-3-319-90572-3
  • Series Print ISSN 1421-1750
  • Series Online ISSN 2374-0280
  • Buy this book on publisher's site