Riesz fractional gradient functionals defined on partitions: nonlocal-to-local variational limits

This paper addresses the asymptotics of functionals with linear growth depending on the Riesz $s$-fractional gradient on piecewise constant functions. We consider a general class of varying energy densities and, as $s\to 1$, we characterize their local limiting functionals in the sense of $\mathrm{\Gamma }$-convergence.