Équations aux dérivées partielles

Sub-Riemannian soap bubbles

"What is the shape of a soap bubble?" At a first sight, the answer seems very easy to guess: soap bubbles are spheres. The physical reason for it is that their surface is minimal among all other forms containing the same volume. Nonetheless, from a mathematical point of view, to formalize and answer this question, very deep concepts are needed.

After an introduction to the soap bubbles problems in the Euclidean setting, in this seminar we focus on anisotropic soap bubble problems. This amounts to find the best configuration of $m\in \mathbb N$ regions in the plane enclosing given volumes, in order to minimize their total perimeter, where perimeter and volume are defined by two different anisotropic densities. The particular structure of such densities is inspired by a subriemannian model, called the Grushin plane, that will be presented. We discuss existence of solutions to the minimal partition problem (ongoing work with Aldo Pratelli - Università di Pisa and Giorgio Stefani - SNS, Pisa) and we conclude by analyzing the particular choices of volume and perimeter associated with the Grushin plane. In this framework we present our candidate solutions for the double bubble problem (joint work with Giorgio Stefani).