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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Elliptic Systems of Phase Transition Type (Hardcover, 1st ed. 2018)
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Elliptic Systems of Phase Transition Type (Hardcover, 1st ed. 2018)
Series: Progress in Nonlinear Differential Equations and Their Applications, 91
Expected to ship within 10 - 15 working days
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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
Cabre (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei
(counterexample for 9d and above). This book extends, in various
ways, the Caffarelli-Cordoba 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.
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