Rafael de la Llave, Enrico Valdinoci
Multiplicity results for interfaces of
Ginzburg-Landau-Allen-Cahn equations
in periodic media
(435K, PostScript)
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.