Francesca Gladiali, Università degli Studi di Sassari, Solutions of the torsion problem with many critical points in Riemannian manifolds, Friday, December 13, 2024, time 11:00, Aula Seminari - III Piano
Abstract:Abstract:
Given a complete Riemannian manifold (M,g) I will prove that, for any point p in M and for any integer n>2, there exists a family of sets containing p and a family of solutions to the torsion problem that admits at least n maximum points. Moreover the domains are close to be convex (in a suitable sense). The proof relies on similar results in R^d, for d>3. The talk is based on past and ongoing results involving M. Grossi and A. Enciso.

Filippo Dell'Oro, Politecnico di Milano, Vanishing viscosity limit for the compressible Navier-Stokes equations with non-linear density dependent viscosities, Friday, November 29, 2024, time 11:00, Aula seminari - III Piano
Abstract:Abstract:
In a three-dimensional bounded domain, we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero. Joint work with L. Bisconti and M. Caggio.

Marco Pozzetta, Politecnico di Milano, Isoperimetric inequalities on manifolds with curvature bounds, Friday, November 22, 2024, time 11:00, Aula Seminari - III Piano
Abstract:Abstract:
The celebrated Euclidean isoperimetric inequality provides a sharp estimate of the measure of the boundary of subsets of the Euclidean space in terms of their volume. This inequality is also rigid, as equality is achieved only by Euclidean balls. The study of sharp and rigid isoperimetric inequalities on Riemannian manifolds has been an active area of research over the past few decades and is closely connected to curvature bounds. In this introductory talk, we will review some classical and recent isoperimetric inequalities on classes of Riemannian manifolds with curvature bounded below. A possible unified approach to the proofs of such inequalities arises from the sharp concavity properties of the isoperimetric profile function. This latter result was first obtained on compact manifolds by C. Bavard--P. Pansu and S. Gallot in the '80s, and it has recently been extended to the noncompact setting in a joint work with G. Antonelli, E. Pasqualetto, and D. Semola.

Danica Basaric, Politecnico di Milano, On well-posedness of systems arising from fluid dynamics, Friday, November 15, 2024, time 10:00, Aula Seminari - III Piano
Abstract:Abstract:
Well-posedness of systems describing the motion of compressible fluids in the class of strong and weak solutions represents one of the most challenging problems of the modern theory of PDEs. In the first part of the talk, we are going to define and construct a larger class of solutions, the measure-valued and dissipative ones, which unable us to handle the problem of existence for large times and large initial data; we will also discuss the possible advantages of considering this weaker notion of solution when solving some related problems arising from fluid dynamics. In the second part of the talk, we are going to show that, by performing a suitable selection, it is possible to select one "good" solution satisfying the semigroup property, even in the context when the system lacks uniqueness. After showing this procedure in an abstract setting, we will apply it to specific systems, such as the compressible Euler and Navier-Stokes ones.

Riccardo Molinarolo, Università degli Studi del Piemonte Orientale "A. Avogadro", A general integral identity with applications to a reverse Serrin problem, Wednesday, October 30, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
The talk aims to present a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that R. Magnanini and G. Poggesi previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As a first application of this new general differential identity, we prove a quantitative symmetry result for the ``reverse Serrin problem'', which we will introduce. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition. This is a joint work with R. Magnanini and G. Poggesi.

Kazuhiro Ishige, The University of Tokyo, Preservation of concavity properties by the Dirichlet heat flow, Wednesday, October 23, 2024, time 16:00, Aula Seminari - III Piano
Abstract:Abstract:
We characterize concavity properties preserved by the Dirichlet heat flow in convex domains of the Euclidean space. (This is a joint work with Paolo Salani and Asuka Takatsu.) Next, we show that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. (This is a joint work with Asuka Takatsu and Haruto Tokunaga.)