Long-range interactions, phase coexistence, and minimal surfaces.

The theory of phase coexistence is a classical field, with foundational contributions, among the others, from Gibbs, van der Waals, and Lev Landau. The associated problem of phase separation connects to the theory of minimal surfaces, as exemplified by the seminal work of Ennio De Giorgi. In this context, phase coexistence is inherently interdisciplinary, encompassing aspects of physics, materials science, mathematical analysis, and differential geometry.
Recently, significant attention has been directed toward models where surface tension, which governs phase separation, emerges as a macroscopic average of long-range particle interactions. This becomes particularly relevant at molecular scales, where nonlocal interactions are not efficiently averaged out. This framework leads to the study of the nonlocal Allen-Cahn equation and fractional minimal surfaces. We will discuss recent results in these areas and highlight some open problems that remain to be addressed.