Existence and rigidity of the Peierls-Nabarro model for dislocations in high dimensions

TL;DR

This paper proves the existence, uniqueness, and one-dimensional symmetry of dislocation solutions in a high-dimensional nonlocal Peierls-Nabarro model, demonstrating that dislocations are strictly monotonic shear profiles under certain conditions.

Contribution

It establishes the existence, smoothness, and symmetry of minimizers for a reduced scalar nonlocal model of dislocations in any dimension, extending previous results to high-dimensional settings.

Findings

Minimizers of the PN energy exist and are smooth one-dimensional profiles.

These profiles are monotonically convergent to stable states at infinity.

Uniqueness up to translation is proven for minimizers and layer solutions.

Abstract

We focus on existence and rigidity problems of the vectorial Peierls-Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg-Landau equation with an anisotropic positive (if Poisson ratio belongs to $(-1/2,1/3)$) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from $H^{1/2}$ regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement.