Elliptic Systems of Phase Transition Type

Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis

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English

Birkhauser Verlag AG
31 January 2019

Differential calculus & equations; Calculus of variations

Series: Progress in Nonlinear Differential Equations and Their Applications

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.

By: Nicholas D. Alikakos, Giorgio Fusco, Panayotis Smyrnelis
Imprint: Birkhauser Verlag AG
Country of Publication: Switzerland
Edition: 2018 ed.
Volume: 91
Dimensions: Height: 235mm, Width: 155mm,
Weight: 699g
ISBN: 9783319905716
ISBN 10: 3319905716
Series: Progress in Nonlinear Differential Equations and Their Applications
Pages: 343
Publication Date: 31 January 2019
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Introduction.- Connections.- Basics for the PDE System.- The Cut-Off Lemma and a Maximum Principle.- Estimates.- Symmetry and the Vector Allen-Cahn Equation: the Point Group in Rn.- Symmetry and the Vector Allen-Cahn Equation: Crystalline and Other Complex Structures.- Hierarchical Structure - Stratification.- Vector Minimizers in R2.- Radial Solutions of ∆u = c2u.

Reviews for Elliptic Systems of Phase Transition Type

This monograph is a useful reference for experts. It is also good for those who want to learn about recent results and methods and start doing research in this field. (Juha K. Kinnunen, Mathematical Reviews, November, 2019)