Surface tension and similar forces lead to area-minimizing interfaces in some physical phenomena, observable at macroscopic scales. However, this principle of surface area minimization does not uniformly apply across all scales, as the underlying physical energies often vary with scale. For example, describing a soap film as an area-minimizing surface becomes implausible at scales comparable to 5 nanometers, the size of a soap molecule. Similarly, the Allen-Cahn energy (i.e., scalar Ginzburg-Landau) exhibits scale-dependent behavior that mirrors area minimization only at larger scales. The regularity theory for absolute energy-minimizing minimal surfaces has been successfully extended to several scale-dependent models, including Allen-Cahn. Yet, extending these results to all stable configurations, which represent the states observable in nature, poses significant challenges. In the talk, I will discuss the pressing open questions and the latest findings regarding stable phase transitions in 3-dimensional environments.
This video was produced by the University of Münster, as part of the workshop Mathematics Münster Mid-term Conference.
