Lorenzo Brasco, Università degli Studi di Ferrara, Eigenvalues of the $p-$Laplacian on general open sets, Thursday, April 23, 2026, time 14:30, Aula seminari, terzo piano
Abstract:Abstract:
We start by reviewing from a variational point of view the classical spectral theory of the Dirichlet-Laplacian. On a general open set, it is well-known that the spectrum may fail to be purely discrete. We then turn our attention to a nonlinear variant of this problem, by considering the case of the $p-$Laplacian with Dirichlet homogeneous conditions. More precisely, we analyze the minmax levels of the constrained $p-$Dirichlet integral: we show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ``at infinity'', it actually defines an eigenvalue. We also prove a quantitative exponential fall-off at infinity for the relevant eigenfunctions: this can be seen as a generalization of classical \v{S}nol-Simon--type estimates to the nonlinear case. Some of the results presented have been obtained in collaboration with Luca Briani (TUM Monaco), Giovanni Franzina (CNR-IAC) and Francesca Prinari (Pisa).

Giona Veronelli, Università degli Studi di Milano-Bicocca, Old and new Sobolev inequalities on manifolds via the ABP method, Thursday, April 16, 2026, time 14:30 o'clock, Aula seminari, terzo piano
Abstract:Abstract:
In the first part of the talk, we will survey the implementation of the ABP method to prove (sharp) isoperimetric inequalities both in Euclidean spaces and on complete Riemannian manifolds with nonnegative curvature and Euclidean volume growth, as well as on minimal submanifolds therein. In particular we will outline the breakthrough contributions by X. Cabré and S. Brendle. In the second part, we will present a recent observation regarding the method, which allows us to recover classical Sobolev and Michael-Simons inequalities on manifolds, as well as to obtain new ones.

Helena Del Río, University of Granada, Denseness results and Bollobás-type theorems for range strongly exposing operators, Thursday, April 09, 2026, time 14:30, Aula seminari III piano
Abstract:Abstract:
We present a new class of bounded linear operators on Banach spaces called Range strongly exposing operators (RSE, in short), which form a natural intermediate class between classical norm-attaining operators and Bourgain's absolutely strongly exposing operators (1977). In the first part of the talk, several foundational results on the denseness of norm-attaining operators are extended to this new setting. In particular, we improve some classical results by Uhl (1976) and Schachermayer (1983), and get some analogous to Acosta (1999). In the second part, we address Bollobás-type theorems for RSE operators, that is, results that allow us approximating simultaneously an operator and a point where the norm is almost attained, and we show that this property provides new characterisations of uniform convexity and complex-uniform convexity. This talk is based on joint works with Geunsu Choi, Audrey Fovelle, Mingu Jung and Miguel Martín.

Mattia Freguglia, Bocconi, Minimizing and min-max Yamabe metrics on conical manifolds, Thursday, April 02, 2026, time 14:30, Aula seminari, terzo piano
Abstract:Abstract:
We discuss the existence of Yamabe metrics on conical manifolds with Ricci-flat tangent cones at singular points. We prove an analogue of Aubin’s classical result, obtaining solutions as minimizers of the Yamabe quotient. In contrast to the smooth case, when this condition fails, minimizers may not exist. In dimension four, and in the presence of at least two Z/2Z-orbifold points, we still obtain solutions via a min-max variational scheme. Based on joint works with Andrea Malchiodi (SNS) and Francesco Malizia (SNS).

Giulio Schimperna, Università di Pavia, Some recent results on the so-called "Cahn-Hilliard-Keller-Segel" system, Thursday, March 19, 2026, time 14:30, Aula Seminari, terzo piano
Abstract:Abstract:
In this talk we will present some mathematical results regarding the so-called "Cahn-Hilliard-Keller-Segel'' system. This is a recently proposed model which couples the Cahn-Hilliard system for phase separation with a further equation describing the evolution of an additional variable $\sigma$. The main application of the model refers to tumor growth processes, in which the phase variable $\varphi$ represents the local proportion of active cancer cells, whereas $\sigma$ denotes the concentration of a chemical substance (for instance a nutrient or a drug) affecting the evolution of the tumor. In this setting, the equation for $\sigma$ may be characterized by a quadratic cross-diffusion term similar to that occurring in the Keller-Segel model for chemotaxis. In the talk we will discuss about existence, uniqueness and regularity of several classes of solutions ("weak", "strong" and "entropic") under various assumptions on the mass and nutrient source terms occurring in the system; in a specific situation we will also analyze the long-time behavior of solutions under the perspective of infinite-dimensional dynamical systems.

Giacomo Sodini, TU Wien, Sobolev functions on spaces of measures and applications, Thursday, March 05, 2026, time 14:30, Aula seminari, III piano
Abstract:Abstract:
After a brief introduction to the Wasserstein and Hellinger-Kantorovich distances on the space of (probability) measures, we discuss two approaches to the definition of differentiable functions on spaces of measures. Provided a suitable (and in a way canonical) reference measure is chosen, we show that the two approaches coincide. Finally, we discuss some possible applications such as the study of HJB equations and stochastic processes on spaces of measures.