Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

Rafael de la Llave, Enrico Valdinoci

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way. We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function). We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties. We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.

Original languageEnglish
Pages (from-to)379-426
Number of pages48
JournalAdvances in Mathematics
Volume215
Issue number1
DOIs
Publication statusPublished - 20 Oct 2007
Externally publishedYes

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Allen-Cahn Equation
Periodic Media
Ginzburg-Landau Equation
Multiplicity Results
Minimizer
Monotonicity
Integer
Phase Coexistence
Variational Model
Double-well Potential
Minimal Solution
Lamination
Kinetics
Term
Graph in graph theory

Cite this

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abstract = "The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way. We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function). We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties. We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.",
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Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media. / de la Llave, Rafael; Valdinoci, Enrico.

In: Advances in Mathematics, Vol. 215, No. 1, 20.10.2007, p. 379-426.

Research output: Contribution to journalArticle

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