Tuesday, 5 June 2018,

* 3 p.m. (sharp),**Dr. Giorgio Saracco**, Università degli Studi di Pavia

at the conference room of IMATI-CNR in Pavia, will give a lecture titled:

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

* 4 p.m. (sharp)**Dr. Stefano Almi,** TUM Monaco

will give a lecture titled:

Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks

as part of the Applied Mathematics Seminar (IMATI-CNR e Dipartimento di Matematica, Pavia).

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Abstract Dr. Saracco:

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)

Abstract Dr. Almi:

In this talk we present some results regarding the singular behavior of the displacement u of a linearly elastic body in dimension 2 close to the tip of a smooth crack. In particular, we will show that u is the sum of an H2-function and of a linear combination of two singular functions whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. We will then apply the above splitting to explicitly compute the derivative of the elastic energy of the system with respect to an infinitesimal fracture elongation, enlightening its dependence on the stress intensity factors. This is a joint work with Dr. Ilaria Lucardesi.