We prove that minimizers of fractional Gagliardo seminorm, among piecewise affine functions defined on the real line with two given slopes (suitably prescribing the length scale of the oscillations) are periodic.
Our results have relevant applications to the van der Merwe theory of misfit dislocations at semi-coherent straight interfaces: We prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one.