Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.