Vittorino Pata, Politecnico di Milano, Weak Solutions of Linear Differential Equations in Hilbert Spaces, Thursday, May 29, 2025, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
We address the well-posedness of weak solutions for a general linear evolution problem on a separable Hilbert space. For this classical problem there is a well-known challenge of obtaining a priori estimates, as a constructed weak solution may not be regular enough to be utilized as a test function. This issue presents an obstacle for obtaining uniqueness and continuous dependence of solutions. When formal energy estimates ara available, we provide a general notion of weak solution and, through a straightforward observation, obtain that arbitrary weak solutions have additional time regularity and obey an a priori estimate. This yields weak well-posedness. Our result rests upon a central hypothesis asserting the existence of a "good" Galerkin basis for the construction of a weak solution. A posteriori, a strongly continuous semigroup may be obtained for weak solutions, and by uniqueness, weak and semigroup solutions are equivalent.

Claudia Bucur, Università degli Studi di Milano Statale, S-minimal functions: existence and continuity, Thursday, May 08, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
We discuss a nonlocal fractional problem that serves as a nonlocal counterpart of the classical problem of functions of least gradient. We show how we obtain the existence of minimizers by using their connection to nonlocal minimal sets. Additionally, we discuss the continuity of these minimizers and a weak formulation of the problem. The results presented are obtained in collaboration with S. Dipierro, L. Lombardini, J. Mazón and E. Valdinoci.

Nicolò De Ponti, Politecnico di Milano, Lipschitz smoothing heat semigroup and functional-geometric inequalities, Thursday, April 10, 2025, time 15:15, Aula seminari MOX - VI piano
Abstract:Abstract:
In this seminar, we present several functional and geometric inequalities, including a Buser-type inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and various estimates involving the 1-Wasserstein distance. Emphasis will be placed on the proof strategy rather than the results, highlighting how all the inequalities are derived using heat semigroup techniques through a key property: a Lipschitz smoothing estimate. This estimate will be presented in detail, and we will show its validity across a broad class of spaces, including Riemannian manifolds with Ricci curvature lower bounds and various sub-Riemannian structures. The seminar is based on joint work with G. Stefani.

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).