**9:00 - 9:45 Ilaria Lucardesi ** (Université de Lorraine)

* On Blaschke-Santalo' diagrams involving the torsional rigidity*

and the first eigenvalue of the Dirichlet Laplacian.

A Blaschke-Santalo' diagram is the range of a vector shape functional (F_{1},F_{2}) in R^{2}.

The determination of such attainable set amounts to completely characterize the relation between F_{1} and F_{2}.
In this talk I will present some recent results obtained in collaboration with D. Zucco, in the case of F_{1} the first Dirichlet eigenvalue and F_{2} the inverse of the torsional rigidity, defined on convex shapes with unit volume, and, as a variant, on convex sets with volume at most 1.
The study led us to address some very deep questions, whose answers are still open problems: in the last part of the talk, I will list them, together with our conjectures.

**9:45 - 10:30 Ekaterina Mukoseeva ** (SISSA)

* Minimality of the ball for a model of charged liquid droplets *

We study minimizers for a variational model describing the shape of charged liquid
droplets. The surface tension forces the particles to stay together whereas the electric
charge causes repulsion. Thus, one expects a certain transition to happen when increasing the charge. This heuristic is indeed confirmed by the experiments first conducted
by Zeleny in the beginning of the previous century. A spherical droplet, when exposed
to an electric field, remains stable until the charge reaches a certain critical value. The
droplet then starts developing singularities.

There are several models trying to capture this phenomenon. We work with the variational model proposed by Muratov and Novaga, as the most commonly used Rayleigh’s
one is ill-posed. Using the recent regularity result by De Philippis, Hirsch, and Vescovo
we are able to prove that the only minimizers in the case of small charge are balls.

This is a joint work with Giulia Vescovo.

**11:00 - 11:45 Baptiste Trey ** (Université Grenoble Alpes)

* Lipschitz regularity of the eigenfunctions on optimal sets minimizing the sum of the k first eigenvalues for an operator with variable coefficients *

The first part of this talk will be addressed to the problem of minimizing the first eigenvalue of the Dirichlet Laplacian with drift in a box. If the drift is the gradient of a function, we can prove a regularity result for the optimal shapes. In a second time, we will consider the minimization of the k first eigenvalues for an operator in divergence form. We can then prove the first step in the regularity theory, that is, the Lipschitz continuity of the eigenfunctions.

**11:45 - 12:30 Dario Mazzoleni ** (Catholic University of Brescia)

* On principal frequencies, volume and inradius in convex sets *

We provide a sharp double-sided estimate for Poincare-Sobolev constants on a convex set, in terms
of its inradius and N−dimensional measure. Our results extend and unify previous works by Hersch
and Protter (for the first eigenvalue) and of Makai, Polya and Szego (for the torsional rigidity), by
means of a single proof. This is a joint work with Lorenzo Brasco (Ferrara).

**12:30 - 13:15 Aldo Pratelli ** (Università di Pisa)

* On the minimality of balls for small volume for the Riesz
functional *

In the last decades, and particularly in the last years, there
has been a lot of effort to study the problem of minimizing the Riesz
functional among sets of given volume in R^{N}. The Riesz functional of a
set is given by the sum of its perimeter and the double integral of a
repulsive potential of the form |y-x|^{p} where p is a negative number
between 0 and -N. While it is more or less obvious that for small
volumes the minimizer should look like a ball, and for big volumes the
mass should be spread away, a remarkable known result is that minimizers
are exactly balls for volume small enoungh. We will describe the problem
and present a new proof of this result, which is valid in a more general
call of functionals which in particular contains the negative powers.