An overview on the Cheeger problem

Abstract

This is an expository article presenting various results about the Cheeger problem and its connections with other relevant mathematical problems. In section 2 we synthetically describe three mathematical problems (eigenvalue estimates, prescribed mean curvature equation, ROF model for image segmentation) focusing on their link with the Cheeger problem. Then, a selection of classical as well as more recent results obtained in collaboration with Aldo Pratelli will be described in the remaining sections.

This article is an extended version of a presentation given at the ERC workshop "New trends in shape optimization" held in Erlangen on Sept. 23-27, 2013