05-225 Rafael de la Llave, Enrico Valdinoci

Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media (435K, PostScript) Jun 23, 05

Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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 at least two equilibrium solutions, which are asymptotic to the pure phases but are separated by an interface. The width of the interface is bounded above by a universal constant and these solutions satisfy some monotonicity properties with respect to integer translations. Also, such solutions approach the pure phases exponentially fast. 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 irrationally oriented planes. In this case, 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 and enjoy monotonicity properties with respect to integer translations.

Files: 05-225.src( desc , 05-225.ps )