Interface minimization of planar networks of branched interfaces

The concept of paired calibrations due to Lawlor and Morgan (Pacific J.\ Math., 166, 1994) provides a particularly elegant tool for proving global area minimization of a network of branched interfaces as appearing in multiphase materials. In this talk, I will present a result on local interface area minimization for a class of stationary points of the interface energy which, due to Kinderlehrer and Liu (Math.\ Models Methods Appl.\ Sci., 11, 2001), occur as long-time asymptotic limits of multiphase mean curvature flow. This class contains continuous one-parameter families of stationary points which are not global minimizers. In particular, paired calibrations do not exist for those and consequently, minimality properties for such networks remained an open problem. Our result is based on a new concept of paired local calibrations allowing to overcome the difficulties associated with the above mentioned degeneracy of the energy landscape. This is joint work with Julian Fischer, Tim Laux and Theresa Simon (arXiv:2212.11840).