Nonlocal phase transitions in homogeneous and periodic media

TL;DR

This paper studies nonlocal phase transition models balancing double-well potential energy with fractional elastic energy, providing asymptotic analysis, flatness, rigidity results, and constructing special minimizers and orbits in periodic media.

Contribution

It introduces new asymptotic and geometric results for nonlocal phase transitions, including planelike minimizers and symbolic dynamics in periodic media.

Findings

Gamma-convergence and energy bounds established

Existence of planelike minimizers in periodic media

Construction of orbits with symbolic dynamics

Abstract

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as $\Gamma$-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media. Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.