Non-local Interfacial Energies in Grain Boundaries via Dirichlet-to-Neumann Map and Energetic scaling

This paper deals with the variational analysis of edge dislocations arising in small-angle grain boundaries (Read--Shockley) and semi-coherent interfaces (van der Merwe). Using the Nabarro--Peierls approach to model edge dislocations, we reduce the two-dimensional isotropic Navier--Lamé system governing the linearized elastic energy to a one-dimensional non-local energy on the slip interface through a Dirichlet-to-Neumann map. We prove that the resulting boundary energy coincides, up to a multiplicative constant, with the fractional \(H^{\frac12}\)-Gagliardo seminorm of the displacement jump along the interface. Within this framework, we establish the existence of a unique global minimizer by combining our dimensional-reduction result with the characterization of De Luca, Goldman, and Ponsiglione. Indeed, we show that the optimal defect configuration is uniquely attained, up to translations, by an equispaced periodic array. Finally, we provide a physical rescaling that recovers the classical logarithmic energy regimes of both the Read--Shockley and van der Merwe models.