Edoardo Mainini, Università di Genova, Gradient flows: from diffusion to Keller-Segel models, Tuesday, May 19, 2026, time 15:15, Aula seminari, terzo piano
Abstract:Abstract:
We discuss the gradient flow approach to aggregation-diffusion models of Keller-Segel type. We prove regularizing effect and sharp hypercontractivity estimates. As a consequence, we deduce several properties such as existence of solutions with measure data, energy dissipation identity, evolution variational inequalities and uniqueness, thus establishing a full parallel with the theory of gradient flows for diffusion models and geodesically convex functionals. 

Camilla Polvara, Sapienza Università di Roma, Critical Neumann problem in cones: bifurcation, stability, and symmetry breaking, Thursday, May 14, 2026, time 14:30, Aula seminari, terzo piano
Abstract:Abstract:
We consider the critical Neumann problem in cones. We prove that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones, which is defined through the first Neumann eigenvalue of the Laplace Beltrami operator on the domain D on the unit sphere, which spans the cone. This immediately implies a symmetry breaking result for the minimizers of the Sobolev inequality. Actually, a bifurcation result from the standard bubbles can be proved. We also present a quantitative Sobolev inequality of Bianchi-Egnell type, which holds in any cone, even if the minimizers are not the standard bubbles. These results are contained in joint works with G. Ciraolo, F. Pacella, and L. Provenzano.

Alberto Maione, Politecnico di Milano, H-compactness for nonlocal linear operators in fractional divergence form, Thursday, May 07, 2026, time 14:30, Aula seminari, terzo piano
Abstract:Abstract:
In this talk, we present the mathematical theory of the homogenisation of composite materials, from its origins in the 1970s to some recent applications. In particular, we discuss a new H-compactness result for possibly non-symmetric and nonlocal linear operators in fractional divergence form. In the second part, we focus on symmetric operators and show that the H-convergence is equivalent to the Gamma-convergence of the associated nonlocal energies. This research is carried out in collaboration with Maicol Caponi (University of L'Aquila) and Alessandro Carbotti (University of Salento).

Peter Schlosser, Graz University of Technology, Institute for Applied Mathematics, Fractional Powers of Vector Operators and the Non-local Fourier Diffusion Law, Tuesday, April 28, 2026, time 14:30, Aula seminari MOX, VI piano
Abstract:Abstract:
Abstract. We investigate fractional powers of the generalized gradient operator with non-constant coefficients, formulated in the Clifford algebra setting. The vector nature of this operator makes classical complex spectral theory inadequate, and the analysis must instead be carried out within the S-spectrum framework. A fundamental obstruction arises from the fact that fractional powers are not defined on the negative real line, which seriously complicates their construction in this setting. Extending earlier work on bisectorial vector operators, we show that previously established weak solutions are in fact strong solutions, and we prove injectivity of the gradient operator. We then introduce a novel approach that rigorously circumvents the negative real line obstruction, yielding a precise definition of the fractional powers together with their main operator-theoretic properties. As an application, we derive a non-local Fourier diffusion law governed by these fractional operators. ----------------------- This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

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.