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### The Cheeger problem and an application to the (constant) Prescribed Mean Curvature problem

##### Dr. Giorgio Saracco, Università degli Studi di Pavia

Sala conferenze IMATI-CNR, Pavia - Tuesday, June 5, 2018 h.15:00

**Abstract. ** Given an open, bounded set Omega, one defines its Cheeger constant, h(Omega) as the infimum of the ratio perimeter over volume among all of its subsets. Evaluating h(Omega) and finding the sets E that attain such a minimum is known as the Cheeger problem.

There are many possible motivations to study such a problem as the constant h(Omega) and minimizers of the ratio play a major role in different areas. In particular we shall discuss the connection with the (constant) prescribed mean curvature problem giving a characterization of existence and uniqueness of solutions in terms of the Cheeger problem.

It will be clear that being able to compute h(Omega) and knowing who the minimizers are is of interest. In general though these are difficult tasks, even in the planar case. We shall show that for a class of a Jordan domains there is a structure theorem for minimizers. On top of that, the so-called inner Cheeger formula holds and this allows to compute the exact value of h(Omega).

These results have been obtained in collaboration with G.P. Leonardi (Università di Modena e Reggio-Emilia, ITALY) and R. Neumayer (Northwestern University, USA)

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