Isoperimetric planar tilings with unequal cells

In this seminar, we consider an isoperimetric problem for planar tilings with possibly unequal repeating cells. We present general existence and regularity results, and we study the classification of planar isoperimetric double tilings, namely tilings with two repeating cells of minimal perimeter. In this case, we explicitly determine the associated energy profile and provide a complete description of the phase transitions. We also comment on possible extensions and discuss some open problems. This is based on joint work with M. Novaga and E. Paolini.