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.
