Heteroclinic connections for nonlocal equations

TL;DR

This paper constructs heteroclinic orbits for a strongly nonlocal integro-differential equation using variational methods, renormalized energy, and free boundary theory, with applications to crystal dislocation models.

Contribution

It introduces a novel approach combining variational, perturbation, and free boundary techniques to analyze nonlocal equations with infinite energy.

Findings

Existence of heteroclinic orbits in nonlocal equations

Development of a renormalized energy functional

Application to crystal dislocation models

Abstract

We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.