Past talks 2025/26
Alessandro Carbotti (Università del Salento)
November 27th, 2025
@ 14:30, Seminar room (2nd floor), Alan Turing Building
Title: H-compactness for nonlocal linear operators in fractional divergence form
Abstract: In this seminar we address the problem of the H-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved.
If symmetry is also assumed, we extend the equivalence between the H-convergence of the operators and the $\Gamma$-convergence of the associated energies. If time allows we will talk about further generalizations. The seminar is based on a joint work with M. Caponi (Univaq) and A. Maione (CRM).
Title: On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3
Abstract: The purpose of this talk is to present the article [1], in which we establish two results: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in R^3 with initial data (u_0,b_0) of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference u_0-b_0. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field b under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.
[1] R. Lucà, C. Peña: "On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3". Preprint: arXiv:2505.09340.
Stefano Lisini (Università degli Studi di Pavia)
November 6th, 2025
@ 14:30, Seminar room (2nd floor), Alan Turing Building
Title: Existence of gradient flow solutions for fractional thin film equations with aggregation on convex domains
Abstract: I will show a gradient flow structure for a family of fractional thin film equations with linear mobility and a reaction term. The problem is posed in a bounded convex domain with the homogeneous Neumann boundary condition. Existence and properties of weak solutions will be illustrated.
Alba Lia Masiello (INdAM & Università degli Studi di Napoli "Federico II")
October 30th, 2025
@ 14:30, Seminar room (2nd floor), Alan Turing Building
Title: Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results
Abstract: This is a joint work with Nunzia Gavitone, Gloria Paoli and Giorgio Poggesi. It is well known that there is a deep connection between Serrin’s symmetry result – dealing with overdetermined problems involving the Laplacian – and the celebrated Alexandrov’s Soap Bubble Theorem (SBT) – stating that, if the mean curvature H of the boundary of a smooth bounded connected open set Ω is constant, then Ω must be a ball. We want to extend the study of such a connection to the broader case of overdetermined problems for Hessian operators and constant higher order mean curvature boundaries. Our analysis will not only provide new proofs of the higher order SBT (originally established by Ros in [2]) and of the symmetry for overdetermined Serrin-type problems for Hessian equations (originally established by Brandolini, Nitsch, Salani, and Trombetti in [1]), but also bring several benefits, including new interesting symmetry results and quantitative stability estimates.
[1] B. Brandolini, C. Nitsch, P. Salani, and C. Trombetti. On the stability of the Serrin problem. J. Differential Equations, 245(6):1566–1583, 2008.
[2] A. Ros. Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana, 3(3-4):447–453, 1987.
Francesco Nobili (Università di Pisa)
September 25th, 2025
@ 14:30, Seminar room (2nd floor), Alan Turing Building
Title: Isoperimetric planar Tilings with unequal cells
Abstract: In this talk, we consider an isoperimetric problem for periodic planar Tilings allowing for unequal repeating cells. We discuss general existence and regularity results and we study classification results for double Tilings, i.e. Tilings with two repeating cells. In this case, we explicitly compute the associated energy profile and we give a complete description of the phase transitions. Based on joint works with M. Novaga and E. Paolini.
Title: Discrete and continuum models of robust biological transportation networks
Abstract: We study a discrete model for formation and adaptation of biological transport networks. The model consists of an energy consumption function constrained by a linear system on a graph. We discuss how structural properties of the optimal network patterns, like sparsity and (non)existence of loops, depend on the convexity/concavity of the metabolic part of the energy functional. We then introduce robustness of the network in terms of algebraic connectivity of the graph and explain its impact on the network structure. Passing to the continuum limit as the number of edges and nodes of the graph tends to infinity, we recover a nonlinear system of PDEs. This elliptic-parabolic system consists of a Darcy's type equation for the pressure field and a reaction-diffusion equation for the network conductance. We explain how the robustness property is reflected on the level of the PDE description. We give both analytical results and systematic numerical simulations for the PDE system, providing interesting insights into the mechanisms of network formation and adaption in biological context.