Nonlocal Phase Transitions with Singular Heterogeneous Kernels

In this paper the study of a non-local Cahn-Hilliard-type singularly perturbed family of functionals is undertaken, generalizing known results by Alberti & Bellettini. The kernels considered include those leading to Gagliardo seminorms for fractional Sobolev spaces. The limit energy is computed via $\mathrm{\Gamma }$-convergence and shown to be an anisotropic surface energy on the interface between the two phases.