Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media

TL;DR

This paper constructs planelike solutions for nonlocal phase transition energies in periodic media, showing their interfaces are close to hyperplanes and establishing the existence of nonlocal minimal surfaces with bounded oscillations.

Contribution

It introduces a method to construct planelike minimizers in nonlocal Ginzburg-Landau energies within periodic structures, linking interface geometry to energy estimates.

Findings

Interfaces are within a bounded distance from hyperplanes.

Existence of planelike nonlocal minimal surfaces in periodic media.

Energy of minimizers is controlled by one-dimensional transition layers.

Abstract

We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane. These solutions are either periodic or quasiperiodic, depending on the rational dependency of the normal direction to the reference hyperplane. Remarkably, the oscillations of the interfaces with respect to the reference hyperplane are bounded by a universal constant times the periodicity scale of the medium. This geometric property allows us to establish, in the limit, the existence of planelike nonlocal minimal surfaces in a periodic structure. The proofs rely on new optimal density and energy estimates. In particular, roughly speaking, the energy of phase transition minimizers is controlled, both from above and below, by the energy of one-dimensional transition layers.