Various Mathematical Questions Arising in Phase Separation Problem

The Modica-Mortola functional of phase transitions is well-known to Gamma converge to the hypersurface area functional as the small parameter tends to zero. It is now known that the control of mean curvature-like quantity is sufficient for the convergence, and more subtle questions can be asked about its property. I review some known results along with open problems and discuss related problems such as Allen-Cahn equation with or without coupled terms, Cahn-Hilliard equation and a functional motivated by the large deviation theory. Some of the questions are of technical and others of intrinsic geometric nature in the setting of geometric measure theory.