28 jan 2025 — 11:30 [open in google calendar]
Trento
Abstract.
I would like to present the results obtained in three recent papers obtained in collaboration with Novaga-Stepanov-Tortorelli, Novaga and Novaga-Nobili about three different kinds of infinite isoperimetric partitions. An isoperimetric partition is a way to subdivide a space in many different regions with prescribed measure so that the common interface between the regions has the smallest possible measure. The three proposed examples are "infinite" in three somewhat different ways.
In the first case we consider infinitely many regions which become smaller and smaller so that the total measure is finite. In this case we have some examples and a very weak regularity result in the planar case.
In the second case we consider a finite (small) number of regions but some of the regions might have infinite measure: we are able to reduce this problem to the usual isoperimetric problem for bubble clusters and hence obtain a description of the isoperimetric partitions in the cases when the corresponding problem for bubble clusters has been solved.
In the third case (which is a work in progress) we consider infinite periodic tilings of the space with a finite number of tiles with prescribed measure: we are able to find the solution in the case of two tiles (with any given prescribed areas) in the plane.