Lauren Conger, California Institute of Technology (Caltech), Monotonicity of Coupled Multispecies Wasserstein-2 Gradient Flows, Wednesday, November 26, 2025, time 11:15, Aula Seminari - III Piano
Abstract:Abstract:
We present a notion of $\lambda$-monotonicity for an $n$-species system of PDEs governed by flow dynamics, extending monotonicity in Banach spaces to the Wasserstein-2 metric space. We show that monotonicity implies the existence of and convergence to a unique steady state. In the special setting of Wasserstein-2 gradient descent of different energies for each species, we prove convergence to the unique Nash equilibrium of the associated energies and discuss the relationship between monotonicity and displacement convexity. This extends known zero-sum (min-max) results in infinite-dimensional game theory to the general-sum setting. We provide examples of monotone coupled gradient flow systems, including cross-diffusion, nonlocal interaction, and linear and nonlinear diffusion. Numerically, we demonstrate convergence of a four-player economic model for market competition, and an optimal transport problem. This is joint work with Ricardo Baptista, Franca Hoffmann, Eric Mazumdar, and Lillian Ratliff.

Elsa Maria Marchini, Politecnico di Milano, On some optimal control problems for moving sets, Thursday, November 20, 2025, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
The talk is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a plane region bounded by geographical barriers. The "contaminated region" is a set moving in the plane, which we would like to shrink as much as possible. To control the evolution of this set, we assign the velocity in the inward normal direction at every boundary point. Three main problems are studied: existence of an admissible strategy which eradicates the contamination in finite time, optimal strategies that achieve eradication in minimum time, strategies that minimize the average area of the contaminated set on a given time interval. For these optimization problems, that we like to see as “time dependent isoperimetric problems”, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions are explicitly constructed in a number of cases. Coauthors: Alberto Bressan and Vasile Staicu

Gioacchino Antonelli , University of Notre Dame, Connected sum of manifolds with spectral Ricci lower bounds, Monday, October 20, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
Let $n>2, \gamma>(n-1)/(n-2)$, and $\lambda$ a real number. Let $\mathrm{Ric}$ denote the lowest eigenvalue of the Ricci tensor. I will show that if $M$ and $N$ are two smooth n-dimensional manifolds that admit a complete Riemannian metric satisfying $-\gamma\Delta+\mathrm{Ric}>\lambda$, then their connected sum also admits a metric with the same property. The construction is geometrically similar to a Gromov-Lawson tunnel, and the range $\gamma>(n-1)/(n-2)$ is sharp for this result to hold. Join work with K. Xu.

David Ruiz, Universidad de Granada, Compactly supported solutions to the stationary 2D Euler equations with noncircular streamlines, Thursday, October 09, 2025, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of this type of solutions has been an open problem until the result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have also been found in the literature, but even the $C^1$ case was left open. In this talk we construct such solutions with regularity $C^k$, for any fixed $k$ given. For the proof, we look for stream functions which are solutions to non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a conveniently constructed 1-parameter family of radial solutions. The linearized operator turns out to be critically singular, and is defined in anisotropic Banach spaces. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).

Patrik Knopf, Università di Regensburg, Nonlocal-to-local convergence of convolution operators and some applications, Wednesday, October 08, 2025, time 11:15, Aula Seminari - III Piano
Abstract:Abstract:
The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn-Hilliard equation), especially when a desired local differential operator cannot be derived by microscopic laws. The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. We discuss the Cahn-Hilliard equation as well as a Navier-Stokes-Cahn-Hilliard model as possible applications. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic). In an ongoing collaboration (joint work with Helmut Abels and Christoph Hurm), we intend to establish strong nonlocal-to-local convergence results with convergence rates for anisotropic kernels satisfying lower regularity assumptions. These results can, for example, be applied to nonlocal phase-field models such as the anisotropic Cahn-Hilliard equation.

Guozhen Lu, University of Connecticut, Optimal stability for Hardy-Littlewood-Sobolev and fractional Sobolev inequalities, Friday, September 05, 2025, time 14:15, Aula Seminari - 3° piano
Abstract:Abstract:
In this talk, we will discuss some recent works on sharp stability for some important geometric and functional inequalities. These include asymptotically sharp lower bounds for the stability for the Hardy-Littlewood-Sobolev and fractional Sobolev inequalities in Euclidean spaces, the stability for the Sobolev inequality on the Heisenberg group where the rearrangement inequality is absent. We recently develop a rearrangement-free argument to establish the stability for such Sobolev inequality.