Calculus of Variations and Geometric Measure Theory

A. Braides

An example of non-existence of plane-like minimizers for an almost-periodic Ising system

created by braidesa on 28 Oct 2013
modified on 19 Sep 2014


Published Paper

Inserted: 28 oct 2013
Last Updated: 19 sep 2014

Journal: J. Stat. Phys.
Volume: 157
Pages: 295-302
Year: 2014


The homogenization of ferromagnetic spin systems in deterministic or random environments, as well as in some aperiodic settings, has been carried over in analogy with the homogenization of surface energies. The computation of an effective surface energy for such systems relies on the characterization of those ground states that follow a planar interface, and the related homogenization formulas. For systems with periodic coefficients it has been shown by Caffarelli and de la Llave that the energy of such ground states can be confined on a strip of finite width around a plane (plane-like minimizers). In this paper we show that this is not the case if the coefficients are uniformly almost periodic by giving an explicit two-dimensional example where there is no ground state confined on a strip. In this example the coefficents are the uniform limit of periodic coefficients (with increasing period).

Keywords: Homogenization, spin systems, plane-like minimizers, Ising systems