Program & Abstracts
Program
Tuesday 17th February
09:00 – 09:30
Opening & Registration
11:00 – 11:30
Coffee Break
13:00 – 14:30
Lunch
16:00 – 16:30
Coffee Break
Wednesday 18th February
11:00 – 11:30
Coffee Break
13:00 – 14:30
Lunch
16:00 – 16:30
Coffee Break
17:15 – 17:25
Closing
Abstracts
Maurizio GARRIONE
Recent results on long-term behavior and stability for PDE systems related to suspension bridges
We present some recent results about PDE systems related to the dynam- ics of suspension bridges. Such systems encompass two coupled equations, governing the evolution of the vertical and of the torsional oscillations of the structure. We analyze the long-term behavior of the solutions and their torsional stability, possibly taking into account the presence of damping and forcing terms.
Joint works with Filippo Gazzola and Emanuele Pastorino (Politecnico di Milano).
Alejandro GÁRRIZ
The heat equation in the hyperbolic space. Asymptotic behaviour
We will present some new results about the large-times behaviour of solutions of the heat equation posed in the hyperbolic space $\mathbb{H}^d$, paying special attention to the general case of integrable initial data, improving the results of J.L. Vázquez in [Asymptotic behaviour for the heat equation in hyperbolic space, Commun. Anal. Geom. 2018] and answering the questions therein. For such initial data the asymptotic profiles were not even known. We will manage to provide not only the asymptotic profiles, with some unexpected results, but also the speed of convergence of the solutions thanks to the definition and the study of an Entropy natural to the problem. This is a joint work with J.A. Cañizo and D.A. Marín.
Gabriele GRILLO
Widder theory for the porous medium equation with rough kernels
We consider weighted porous medium equation with rough and inhomogeneous density that may be singular at a point and tends to zero at spatial infinity. No regularity for the density is assumed, therefore forcing us to consider the very weak, i.e. distributional sense for solutions. We first identify an optimal class $X$ of initial measure data that give rise to very weak solution. Then we show that non-negative very weak solutions necessarily admit an initial trace in $X$ at time $t = 0$, and we prove that any two non-negative solutions having the same initial trace are equal. The corresponding theory for the classical (unweighted) equation was established by exploiting various properties that are not available in our weighted setting, such as the continuity of solutions, the explici scale invariance of the equation, Aleksandrov's reflection principle, and the Aronson-Bénilan inequality. We stress that the results are new even for weights which are bounded and bounded away from zero, but are just measurable. Potential methods have a relevant role in our analysis, and the proof of local smoothing effects, of independent interest, are also crucial in our analysis.
This is a joint work with M. Muratori, T. Petitt, N Simonov.
Yuanyuan LI
The $L_p$ chord Minkowski problem for super-critical exponent
In this talk, we consider the $L_p$ chord Minkowski problem, proposed by Lutwak-Xi-Yang-Zhang [CPAM 2024]. We apply a nonlocal Gauss curvature flow and a topological argument to solve this problem for the super-critical exponents. Notably, we provide a simplified argument for the topological part. This is joint work with Shibing Chen (USTC) and Qirui Li (ZJU).
Elsa MARCHINI
Optimal control problems for moving sets with geographical constraints
This talk deals with geometric optimization problems modeling the spatial control of an invasive population within a plane region bounded by geographical barriers. The aim is to shrink as much as possible the "contaminated" region described as a set moving in the plane. To control the evolution of this set, we assign the velocity in the inward normal direction at every boundary point. If no control is applied, the contaminated set expands with unit speed in all directions. Aim of this talk is to provide an extensive analysis of the above problems, in the presence of geographical constraints. Results are presented on existence of an admissible strategy which eradicates the contamination in finite time, on optimal strategies that achieve eradication in minimum time, on strategies that minimize the average area of the contaminated set on a given time interval. Sufficient conditions for optimality and various necessary conditions will be provided and applied to explicitly construct optimal solutions. Further developments will be discussed.
Joint work with Alberto Bressan and Vasile Staicu.
Noema NICOLUSSI
Optimal eigenvalues on a metric graph with densities
Motivated by the notion of conformal eigenvalues for surfaces, we investigate Laplacians on a metric graph with varying mass density. This setting provides a common framework for several well-studied classes of operators: discrete Laplacians, Dirichlet-to-Neumann operators on graphs, and Kirchhoff Laplacians (a.k.a. quantum graph operators). Our main interest is the spectral optimization problem for eigenvalues with respect to the underlying mass density, which turns out to behave rather differently than in the manifold setting. In particular, we discuss the relation of optimal eigenvalues with geometrical properties, including a complete geometric description of the first optimal eigenvalue and a Weyl law.
Based on joint work with Kiyan Naderi (University of Innsbruck).
Giampiero PALATUCCI
The De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusion
I will present some recent results for weak solutions to a wide class of kinetic integral equations, where the diffusion term in velocity is an integro-differential operator having nonnegative kernel of fractional order $s$ in $(0,1)$ with merely measurable coefficients. In particular, I will focus on Harnack-type inequalities for nonnegative weak solutions that does not require the usual a priori boundedness.
The talk is based on a series of papers by Anceschi, Kassmann, Piccinini, Weidner and myself.
Troy PETITT
Rigidity of weighted manifolds via classification results for semilinear equations
We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-Émery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is based on a joint work with G. Ciraolo and A. Farina.
Lei QIN
Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities
It is well known that several variational functionals verify Brunn-Minkowski type inequalities, such as: Torsional rigidity; First Dirichlet eigenvalue of the Laplace; $p$-Capacity and so on. And their solutions of the corresponding boundary value problem (PDE's+boundary conditions), satisfy specific concavity properties. In this talk, I will present my recent work about the first Dirichlet eigenvalue problem to the weighted $p$-Laplace operator in Florence. This is a joint work with professors Andrea Colesanti and Paolo Salani.
Giuseppe SAVARÉ
Variational convergence of metric gradient flows
In this talk we address variational convergence and stability properties of gradient flows in metric spaces. We begin by introducing the notion of metric gradient flow in the sense of Evolution Variational Inequalities (EVI), which provides a robust and intrinsic characterization of dissipative evolutions beyond the Hilbertian setting.
Given a sequence of functionals $(\phi_h)$ defined on a metric space and generating corresponding gradient flows, we study the problem of understanding when and how the associated evolutions converge as the functionals vary. In the classical Hilbert space framework, convergence of gradient flows is closely related to Mosco (i.e. $\Gamma$-convergence with respect to the strong and weak topology) of convex energies and to the convergence of resolvent operators. In a general metric setting, however, several key tools are missing: resolvents may fail to be well defined or contractive, compactness is often unavailable, and no natural weak topology is present.
We present a variational approach to stability that avoids strong coercivity assumptions and relies instead on quantitative error estimates for minimizing movement schemes and their Ekeland relaxations. These estimates provide a robust bridge between discrete variational approximations and continuous EVI flows, and allow one to transfer convergence properties from the energies to the evolutions under suitable variational assumptions.
The results apply to a broad class of metric spaces admitting EVI gradient flows and yield general convergence criteria that extend classical Hilbert space results to genuinely metric frameworks, with applications for instance to Wasserstein and RCD spaces.
Joint work with Matteo Muratori.
Flavia SMARRAZZO
Flux-saturated scalar conservation laws with measure initial data
We study a class of scalar conservation laws with possibly bounded (flux-saturated) nonlinear fluxes and initial data given by finite Radon measures, whose singular part consists of a finite superposition of Dirac deltas. In this setting, solutions preserve initial singularities for positive times, leading naturally to a measure-valued framework.
After introducing the notion of weak and entropy solutions for measure data, we discuss persistence, monotonicity, and continuity properties in time of the singular parts of entropy solutions. We also address the issue of nonuniqueness of entropy solutions in the measure-valued framework and introduce a notion of admissible solutions based on suitable compatibility conditions near the support of the singular part. These conditions can be interpreted as infinite boundary data for the regular part and, combined with the entropic formulation, allow us to recover comparison principles and uniqueness. Finally, we discuss existence results for admissible solutions, obtained either by approximation of the initial data and the flux, or by gluing solutions of suitable singular initial-boundary value problems.
Based on a series of joint papers with Michiel Bertsch, Alberto Tesei, and Andrea Terracina.
Alberto TESEI
Semilinear parabolic equations on metric trees
We discuss some recent results, obtained jointly with Fabio Punzo, concerning the Cauchy-Neumann problem on a regular metric tree for the semilinear heat equation with forcing term of KPP type. Propagation and extinction of solutions, as well as asymptotical speed of propagation, are investigated. Analogies and differences with respect to companion results in the Euclidean or hyperbolic space will be addressed.
Eugenio VECCHI
Sobolev critical problems for local and nonlocal operators
In this talk I will focus on Sobolev critical problems in bounded domains where the operator is given by the superposition $-\Delta \pm (-\Delta)^s$ with $s\in (0,1)$, possibly losing the maximum principles. I will discuss how the presence of the nonlocal term affects the classical results holding for the Laplacian. The talk is based on joint works with S. Biagi, S. Dipierro, and E. Valdnoci.
Bruno VOLZONE
New advances in some anisotropic nonlinear diffusion equations
In this talk we describe several aspects related to the theory of some anisotropic parabolic equations. The anisotropy shown in such equations will appear in the form of porous medium, in the fast and porous medium diffusion regime. In particular, we show the existence of self-similar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The investigation of both models are objects of joint works with F. Feo and J. L. Vázquez.
Marvin WEIDNER
Optimal regularity for kinetic equations in domains
The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. As a first step toward understanding the boundary case, in this talk I will discuss the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov's equation. I will also mention recent results on other boundary conditions such as diffuse reflection and in-flow. This talk is based on joint works with Xavier Ros-Oton and Kyeongbae Kim.
