TY - JOUR

T1 - Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

AU - de la Llave, Rafael

AU - Valdinoci, Enrico

PY - 2007/10/20

Y1 - 2007/10/20

N2 - 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.

AB - 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.

KW - Critical points

KW - Existence and multiplicity results

KW - Ginzburg-Landau-Allen-Cahn equation

KW - Ljusternik-Schnirelmann category

KW - Minimizers

KW - Phase transitions

KW - Plane-like solutions

KW - Qualitative properties of solutions

UR - http://www.scopus.com/inward/record.url?scp=34447507902&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2007.03.013

DO - 10.1016/j.aim.2007.03.013

M3 - Article

AN - SCOPUS:34447507902

VL - 215

SP - 379

EP - 426

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -