Overview
- Can be used as principal or supplementary reading for advanced graduate courses on nonlinear analysis or PDEs
- Features a self-contained exposition of the existence of connections
- Presents a new existence proof of Schatzman's asymmetric double connection solution
- Includes a complete chapter on stable standing and traveling waves for multistable systems based on variational techniques, thus avoiding the more complicated, topological approach
Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 91)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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 theapplied mathematics of materials science.
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Authors and Affiliations
Bibliographic Information
Book Title: Elliptic Systems of Phase Transition Type
Authors: Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
Series Title: Progress in Nonlinear Differential Equations and Their Applications
DOI: https://doi.org/10.1007/978-3-319-90572-3
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Hardcover ISBN: 978-3-319-90571-6Published: 31 January 2019
eBook ISBN: 978-3-319-90572-3Published: 21 January 2019
Series ISSN: 1421-1750
Series E-ISSN: 2374-0280
Edition Number: 1
Number of Pages: XII, 343
Number of Illustrations: 49 b/w illustrations, 10 illustrations in colour
Topics: Partial Differential Equations, Calculus of Variations and Optimal Control; Optimization, Ordinary Differential Equations