Singular kernels, multiscale decomposition of microstructure, and dislocation models

TL;DR

This paper analyzes a dislocation model in crystals with a singular kernel, deriving a sharp-interface limit using $ ext{Gamma}$-convergence, and reveals microstructure properties through analytical techniques.

Contribution

It introduces a new analytical approach for a vector-valued dislocation model with singular kernels, establishing a sharp-interface limit and microstructure characterization.

Findings

Established a sharp-interface limit via $ ext{Gamma}$-convergence.

Proved microstructure is approximately one-dimensional on most scales.

Extended analysis to vector-valued functionals beyond previous scalar cases.

Abstract

We consider a model for dislocations in crystals introduced by Koslowski, Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $\Gamma$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound.