Singular Kernels, Multiscale Decomposition of Microstructure, and Dislocation Models

We consider a model for dislocations in crystals introduced by Koslowski, Cuitino 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, Bouchitte 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 we exploit this property to derive a sharp lower bound.