Past talks 2025/26
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.