Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

We investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions. Through formal derivations, we show that nonlocal diffusion interface problems converge to their classical local counterparts, in the limit of vanishing nonlocality. A central focus of this paper is local/nonlocal interface problems, or interface problems with sharp transitions between local and nonlocal domains. Such problems can be cast as instances of a nonlocal interface problem, with a finite horizon in certain regions and a vanishing horizon in other regions. We derive a local/nonlocal interface problem and utilize conservation principles to obtain local/nonlocal interface conditions. Comparisons between nonlocal, local, and local/nonlocal interface problems are presented, analytically and numerically, with a focus on multiscale aspects of nonlocal models induced by their inherent length scales.