Alberto Farina, Université de Picardie Jules Verne, Monotonicity for solutions to semilinear problems in epigraphs and applications, Thursday, April 03, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when $f$ is a (locally or globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. Also, we answer a question (raised by Berestycki, Caffarelli and Nirenberg) about Serrin's overdetermined problems on epigraphs.

Tamas Titkos, Corvinus University and Rényi Institute., Rigid and non-rigid Wasserstein spaces, Thursday, March 27, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
In recent decades, the theory of optimal transport has advanced rapidly, finding an ever-growing range of applications. The original problem of Monge is to find the cheapest way to transform one probability distribution into another when the cost is proportional to the distance. The most important metric structure that is related to optimal transport is the so-called p-Wasserstein space [denoted by Wp(X)] over the metric space X. The pioneering work of Bertrand and Kloeckner started to explore fundamental geometric features of 2-Wasserstein spaces, including the description of complete geodesics and geodesic rays, determining their different types of ranks, and understanding the structure of their isometry group. In this talk I will focus on isometry groups. A notable and useful property of p-Wasserstein spaces is that X embeds isometrically into Wp(X), moreover an isometry of X induces an isometry of Wp(X) by the push-forward operation. These induced isometries are called trivial isometries, and we say that Wp(X) is isometrically rigid if all its isometries are trivial. The question is: are there non-rigid Wasserstein spaces? What does a non-trivial isometry look like? Until very recently, only a few non-rigid examples were known such as the 2-Wasserstein space over R^n, and the 1-Wasserstein space over [0,1]. In the first part of the talk, I will introduce some key concepts and notation. The main focus will then shift to exploring results concerning both rigidity and non-rigidity in Wasserstein spaces.

Daniele De Gennaro, Department of Decision Sciences and BIDSA, Bocconi University, Milano, Italy, A geometric stability inequality and applications to the stability of geometric flows , Thursday, March 13, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
In this talk I will discuss some results concerning geometric flows. In particular, we will focus on flows with a volume constraint and whose motion is depending on their mean curvature, which (formally) arise as gradient flows for the perimeter functional. After an introduction on the topic, aimed at a general audience, we will discuss a novel geometric inequality, which takes the form of a quantitative Alexandrov theorem, in the periodic setting. We will then show how to use this inequality to prove global existence and to characterize the asymptotic behaviour for some instances of volume-preserving geometric flows. Our results apply to the volume-preserving mean curvature flow, the surface diffusion flow and the Mullins-Sekerka flow. This work is based on a collaboration with Anna Kubin (TU Wien), Andra Kubin (University of Jyväskylä) and Antonia Diana (Sapienza University).

Danica Basaric, Politecnico di Milano, On well-posedness of systems arising from fluid dynamics, Thursday, March 06, 2025, time 15:15, 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.

Ruijun WU, Beijing Institute of Technology, Estimates on the nodal sets of solutions to Dirac equations, Friday, Febraury 14, 2025, time 11:00, Aula Seminari - III Piano
Abstract:Abstract:
Motivated by the various Dirac equations in geometry and physics, we consider the nodal set of solutions to a class of Dirac equations. In contrast to scalar function case, we show that the nodal sets in general has codimension at least two, which strengthen the known results in the smooth setting and confirms a conjecture in spin geometry. Moreover, using a spinorial version of the frequency function, we show that the nodal set can be stratified nicely. This is based on a joint work with A. Malchiodi and W. Borrelli.

Alejandro Fernandez-Jimenez, University of Oxford, Aggregation-diffusion equations with saturation, Friday, January 31, 2025, time 11:00, Aula Seminari MOX - VI piano
Abstract:Abstract:
On this talk we will focus on the family of aggregation-diffusion equations $$ \frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho) \nabla (U'(\rho) + V) \right). $$ Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$. For this problem, we discuss: Existence using a suitable regularised approximation of the problem, we prove that the problem admits an $L^1$-contractive $C_0$-semigroup; $L^1$-local minimisers of the associated free-energy functional in the corresponding class of measures; and the long-time behaviour of the constructed solutions in view of its gradient flow structure. Furthermore, we observe saturation effects leading to "freezing" behavior, i.e. free boundaries at the saturation level. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu. The talk presents joint work with Prof. J.A. Carrillo and Prof. D. Gómez-Castro.