Florian Fischer, University of Bonn, Optimal Poincaré-Hardy inequalities on graphs, Tuesday, July 01, 2025, time 14:15, Aula Seminari III Piano
Abstract:Abstract:
We review a method to obtain optimal Poincaré-Hardy inequalities on the hyperbolic spaces by Berchio, Ganguly and Grillo. Then we show how to transfer the basic idea to the discrete setting. This yields optimal Poincaré-Hardy-type inequalities on model graphs which include fast enough growing trees and anti-trees. Moreover, this method yields optimal weights which are larger outside of a ball than the optimal weights constructed via the Fitzsimmons ratio of the square root of the minimal positive Green's function. Joint work with Christian Rose.

Marco Bramanti, Politecnico di Milano, Stime a priori in spazi di Sobolev per EDP lineari del second'ordine e analisi reale, Thursday, June 26, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
In questo seminario vorrei descrivere un filone di ricerca che ruota attorno alle stime a priori in spazi di Sobolev per soluzioni di equazioni lineari del second'ordine di tipo non variazionale ellittiche, paraboliche, ultraparaboliche, a coefficienti poco regolari (eventualmente discontinui). In particolare presenter\`{o} alcuni risultati ottenuti con Stefano Biagi su una certa classe di operatori di tipo Kolmogorov-Fokker-Planck. I temi che si intrecciano sono: le tecniche di analisi reale e di teoria degli integrali singolari, anche in contesti non Euclidei; la conoscenza di propriet\`{a} fini delle soluzioni fondamentali di certi operatori modello; il ruolo di strutture geometriche (traslazioni, dilatazioni, distanza) adattate a questi operatori modello. Il seminario sar\`{a} il pi\`{u} possibile discorsivo e non tecnico.

Jonas Stange, University of Regensburg, A convective Cahn–Hilliard model with dynamic boundary conditions, Thursday, June 05, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
We consider a general class of convective bulk-surface Cahn–Hilliard systems with singular potentials. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters $K,L\in [0,\infty]$, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. In this talk, I present some recent results on the well-posedness of this system. After briefly recalling the results for regular potentials, we focus on singular potentials. Here, we make use of the Yosida approximation to regularise these potentials, which allows us to apply the results for regular potentials and eventually pass to the limit in this approximation scheme to obtain a global-in-time weak solution. Afterwards, under additional assumptions on the mobility functions, we prove higher regularity estimates for two different classes of velocity fields, and in particular, for those having Leray-type regularity. Finally, exploiting these higher regularity estimates, we can establish separation properties of the phase-fields. This is based on joint work with Andrea Giorgini and Patrik Knopf.

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.

Carlo Nitsch, Università degli Studi di Napoli - Federico II, Improving a Spectral Inequality by Payne, Thursday, May 22, 2025, time 15:15, Aula B.4.4
Abstract:Abstract:
A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne’s original estimate—which is not sharp—can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.

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.