Interface controlled models of phase-boundaries are considered. Evolving phase-boundaries may develop singularities in a finite time. A level set approach developed by Chen-Giga-Goto and Evans-Spruck is explained for nonmathematicians. This method is usefel to track the evolution of the phase-boundary after it experiences singularities. A key mathematical tool is the theory of viscosity solutions which are generalized solutions of the second order degenerate elliptic and parabolic equations. This important theory is also explained without touching technical details.
