Deterministic mean field games without the Hamilton–Jacobi equation: a fully variational approach beyond the separable case

• Antonio Siconolfi (Università degli Studi di Roma "La Sapienza")

Room: Aula Dini In the traditional approach, as well as in the one nowadays usually referred to as Lagrangian, the Hamilton–Jacobi equation, with Hamiltonian given by the convex dual of the Lagrangian cost, plays a crucial role in the analysis of first-order MFG problems. The main reason is that the vector field driving the continuity equation is expressed in terms of Hp(·, ·, −Dv(x, t)), where v is the value function. To make this expression meaningful, some analysis of the HJ equation is required, typically to establish semiconcavity properties of v and, in turn, enough differentiability along the relevant characteristics. On the contrary, we show that a vector field driving the continuity equation can be defined under rather general assumptions on the Lagrangian, solely by exploiting the rep- resentation formula for the value function, without ever using the fact that it solves the corresponding HJ equation. This paves the way for a purely variational analysis of the problem and, in particular, allows us to dispense with the separable structure commonly imposed on Lagrangians in the MFG literature. This is ongoing research in collaboration with Marco Pozza (Unilink).

On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction

• Ariela Briani (Institut Denis Poisson, Université de Tours)

Room: Aula Dini I will describe my recent contribution to the theory of Hamilton-Jacobi equations and optimal control problems posed on networks (see [1]). I will study evolutive Hamilton–Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. I will introduce a notion of (flux-limited) viscosity solution for Hamiltonians H_i=H_i(t,x,p), (i=1,2) that are assumed to be only measurable in t. Moreover the flux limiter,  A=A(t), acting at the junction, is not required to be continuous but only in L-infinity. My first motivation is the optimal control problem studied by P. Cardaliaguet et P. Souganidis in [2], where they investigate how to optimally control traffic flow through a junction by acting only on the junction (i.e. with speed reduction or a traffic light) using a merely measurable flux limiter A(t). In the case of convex Hamiltonians, I prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalizations to the nonconvex case and to more general networks are also discussed. [1] A. Briani, On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction , Preprint (2026), ArXiv https://arxiv.org/abs/2603.04183. [2] P. Cardaliaguet, P. E. Souganidis, An optimal control problem of traffic flow on a junction, preprint, (2023), https://arxiv.org/abs/2312.15418.

Quantitative vanishing viscosity approximation of fully nonlinear, non-convex, Hamilton-Jacobi equations with Hölder data

• Alessandro Goffi (Università di Firenze)

Room: Aula Dini I will discuss new quantitative estimates of the vanishing viscosity process for evolutionary Hamilton-Jacobi PDEs that are neither concave nor convex in the gradient and Hessian entries. I will describe a novel approach that exploits the regularizing properties of sup/inf-convolutions for viscosity solutions combined with the comparison principle. This method provides explicit sharp constants without assuming differentiability properties neither on solutions nor on the Hamiltonian. This is a joint work with Alekos Cecchin (Padova).

Constrained estimation and stochastic filtering

• Hasnaa Zidani (INSA Rouen Normandie)

Room: Aula Dini We study the estimation of the state of a non-smooth dynamics constrained to a bounded convex domain, given noisy observations. The associated value function turns out to be a viscosity sub-solution of an HJB equation with one Neumann-type boundary condition, and a super-solution with a different one — an asymmetry caused by the loss of time-reversibility of sweeping processes, which obstructs uniqueness in general. We recover uniqueness under an inward-pointing drift assumption. We then show, by duality rather than comparison, that this value function arises in the small-noise limit of the corresponding reflected filtering problem, via a Laplace principle.

From Discrete Weak KAM Theory to Computable Aubry–Mather Sets

• Cristian Mendico (Université Bourgogne Europe)

Room: Aula Dini Aubry and Mather sets describe the invariant structures selected by action minimization in Hamiltonian dynamics. Their computation is particularly relevant in celestial mechanics, where resonances and transitions between regular and chaotic motion create intricate dynamical patterns that are difficult to detect using trajectories alone. In this talk, I will present joint work with Fabio Camilli on semi-discrete and fully discrete approximations of these sets. Our approach combines discrete Lax–Oleinik operators with a finite-dimensional variational formulation for minimizing measures. The resulting schemes approximate critical values, weak KAM solutions, and the supports of invariant minimizing measures, while preserving the variational structure of the continuous problem. I will also discuss the numerical challenges created by multiple minimizing components and outline how calibration defects and structure-preserving learning may help identify the variational skeleton of resonant Hamiltonian systems.

MFG with large discount and agent-based models

• Elisa Continelli (Università degli Studi di Padova)

Room: Aula Dini We focus on a class of Mean Field Games with discount. If one assumes that the discount factor is large, these models turn out to behave similarly to agent-based models, where individuals just react to the population distribution according to a given rule. Inspired by a recent work by Bardi and Cardaliaguet, we will elaborate on this connection, showing in particular the uniqueness of equilibria in MFG with large discount, and their converge to certain agent-based models, in a quantitative sense, as the discount factor goes to infinity. Joint work with Marco Cirant.

Hamilton-Jacobi equations on Wasserstein spaces

• Daniela Tonon (Università degli Studi di Padova)

Room: Aula Dini In this talk, we present a unified approach to Hamilton–Jacobi equations on the Wasserstein space of probability measures arising from deterministic and stochastic dynamics. We introduce a viscosity-solution framework encompassing both first-order equations and semilinear equations driven by idiosyncratic noise. The framework relies on a suitable notion of subdifferential designed to ensure comparison and stability. We then discuss the vanishing-viscosity limit for semilinear Hamilton–Jacobi equations, proving convergence to the corresponding first-order equation as the noise intensity vanishes, with an optimal convergence rate. These results provide a PDE-level characterization of the zero-noise transition.

Building on Lax’s Idea: Quantitative Compactness Estimates for Numerical Solutions of First-Order Nonlinear PDEs

• Alessio Basti (Università G. D'Annunzio di Chieti-Pescara)

Room: Aula Dini In my talk, I will discuss recent results on Kolmogorov ε-entropy for numerical approximations of scalar conservation laws and Hamilton-Jacobi equations. The analysis provides quantitative compactness estimates and entropy-transfer principles linking exact and discrete solution sets, yielding an information-theoretic perspective on complexity preservation and the resolution of numerical schemes in the sense introduced by P. D. Lax.

Analysis and approximation of first order Mean Field Games with Hamiltonian measurable-in-time and $\mathcal{C}^{1,1}$ in the gradient variable

• Valentina Coscetti (Università degli Studi di Roma "La Sapienza")

Room: Aula Dini We present a numerical scheme for a class of first-order, non-local Mean Field Games systems with two sources of low regularity: the time-dependent data are merely measurable, and the Hamiltonian is only of class $\mathcal{C}^{1,1}$ with respect to the momentum variable. The approach combines a semi-Lagrangian approximation of the Hamilton-Jacobi-Bellman equation with a Lagrange-Galerkin discretization of the continuity equation. We discuss convergence results and present numerical experiments to illustrate the performance of the method.

SLTK: A Semi-Lagrangian ToolKit for PDEs on Unstructured Grids

• Simone Cacace (Università degli Studi di Roma "La Sapienza")

Room: Aula Dini In this talk, I will present recent advancements in the development of SLTK, a custom research library designed for the numerical solution of PDEs on complex domains. By employing semi-Lagrangian schemes on unstructured grids, SLTK adapts various techniques from computational geometry and computer graphics to the context of partial differential equations. Specifically, I will introduce a generalized point location method for the efficient tracking of characteristics in advection and advection-diffusion problems, including R-tree spatial indexing for the treatment of boundary conditions. Furthermore, I will present a mass-conservative method for the tracking and remapping of mesh elements deformed by a transport field in continuity and Fokker-Planck equations, also introducing an exact geometric folding algorithm for managing flux-type boundary conditions. Finally, I will present some numerical results obtained by applying SLTK to optimal control problems, Mean Field Games, and fluid dynamics on possibly non-convex domains with arbitrary holes. This is a joint work with R. Ferretti (Roma Tre University) and G. Tatafiore (Sapienza University of Rome).

Monotonicity Methods for time-dependent Mean Field Games:

• Diogo A. Gomes (KAUST)

Room: Aula Dini Abstract. We prove the existence of solutions to first-order, local, time-dependent mean-field games with periodic boundary conditions on the d-dimensional torus. Our approach extends the Banach space monotone operator framework, previously developed for the stationary case, to the substantially harder time-dependent setting. We reformulate the MFG system as a variational inequality for a monotone operator and introduce a low-order p-Laplacian regularization that restores coercivity without high-order smoothing. For the regularized problems, existence follows from an abstract monotone operator theorem. We then derive uniform a priori estimates, including energy bounds, higher integrability via the nonlinear adjoint method, and, by passing to the limit using Minty's method, obtain variational inequality solutions. We further show that these solutions are strong solutions to the original MFG system, adapted to the BV framework. Compared to earlier Hilbert space approaches that rely on high-order elliptic regularization, our method operates in natural Banach spaces, yields stronger solutions, and provides a framework better suited to numerical algorithms.

Mean Field Games with state constraints and Grushin type dynamics

• Paola Mannucci (Università degli Studi di Padova)

Room: Aula Dini We consider a class of finite horizon deterministic mean field games with nonlocal coupling where the agents must follow Grushin type dynamics with state constraints. We require some assumptions on the local interplay between the set of state constraints and the dynamics. As a first step, we consider the associated optimal control problem and we establish some properties as: the existence of an optimal trajectory for any starting point $(x,t)$, the closed graph property for the multivalued map which associates to each point $(x,t)$ the set of optimal trajectories starting from that point, the continuity of the value function. Moreover, using a penalization method and the Maximum Principle, we obtain an uniform bound for the optimal controls. This allows us to obtain a comparison principle and that the value function is the unique constrained viscosity solution of the associated HJ equation. Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state). Research project in collaboration with Alessandra Cutri' (Roma Tor Vergata), Claudio Marchi (Universita' di Padova), Nicoletta Tchou (Universite' de Rennes)

Approximation of deterministic mean field games under state constraints

• Francisco J. Silva (Université de Limoges)

Room: Aula Dini In this talk, we study the approximation of equilibria for continuous-time and continuous-space first-order (deterministic) mean field games (Mean Field Games, MFGs) by equilibria of finite MFGs, i.e., games in which the number of time steps and the state space are finite. In the first part, we recall the framework and results obtained in Hadikhanloo–S (2019). In the second part, based on a join work with J. Gianatti and F. Mierez, we extend the analysis to the case where constraints are imposed on the agents’ states. We also present numerical simulations of approximate MFG equilibria in both the unconstrained and constrained cases.

Rate of convergence for singular perturbations of Hamilton-Jacobi equations in unbounded spaces

• Daria Ghilli (Università di Pavia)

Room: Aula Dini We present rate of convergence results for singular perturbations of Hamilton-Jacobi equations in unbounded spaces where the fast operator is linear, uniformly elliptic and has an Ornstein-Uhlenbeck-type drift. Our achievements are the first rate of convergence results without any periodicity assumption. More in detail, in our HJ equation, the slow operator is a fully nonlinear elliptic operator while the source term is assumed only locally Hölder continuous in both fast and slow variables. We obtain several rates of convergence according to the regularity of the source term.

Degenerate second-order PDEs on networks with Kirchhoff conditions

• Olivier Ley (INSA de Rennes)

Room: Aula Dini The aim of this talk is to present results concerning nonlinear degenerate elliptic partial differential equations (PDEs) on networks, with Kirchhoff conditions at the vertices. While elliptic PDEs on networks have long been known, Hamilton-Jacobi equations on networks (the case of totally degenerate PDEs) have been the subject of extensive study over the past decade. In recent work conducted with Guy Barles (Tours) and Erwin Topp (Rio), we obtained results for the general case of degenerate second-order elliptic PDEs.

On the null controllability of a degenerate Fokker–Planck equation with a drift term

• Genni Fragnelli (Università di Siena)

Room: Aula Dini The Fokker–Planck equation describes the time evolution of the probability density function of the velocity for a particle under the influence of drag forces and random forces. In particular, this equation has multiple applications in information theory, graph theory, data science, finance, economics... In one spatial dimension the Fokker–Planck equation for the probability density $p(t,x)$ can be rewritten as \[ p_t(t,x) - (a(t,x)p(t.x))_{xx} + (\mu(t,x)p(t,x))_x= f(t,x) \] where $t\in [0, T]$, $T>$ is fixed and $x \in (0,1)$. In this talk we assume that $a$ is a function degenerating at $x=0$; the purpose is to study the null-controllability of the solution, namely the possibility to drive the solution $p$ to rest completely at time $T$.

Approximation of stable solutions of second order mean field games

• Jules Berry (Université Paris-Saclay)

Room: Aula Dini In this talk, we are interested in the stationary MFG system: \begin{equation} \begin{cases} -\Delta u + H(x,Du) + \lambda u = F[m] & \text{in } \mathbb{T}^d, \\ -\Delta m - \mathrm{div}(m H_p(x,Du)) + \lambda m = \lambda m_0 & \text{in } \mathbb{T}^d, \end{cases} \end{equation} where $u$ is the value function and $m$ the distribution of the players. Our goal is to establish the existence of finite element approximations of the solutions to the MFG ssytem and to obtain the associated error estimates. Our approach consists in reformulating system in the form $F(u,m) = 0$, where $F$ is a nonlinear map defined on a suitably chosen Banach space. In the case where this map is of class $C^1$, we show that the stable solutions, in the sense of Briani-Cardaliaguet, correspond to the regular zeros of $F$, i.e., those for which $dF[u,m]$ is invertible. This property makes it possible to apply the Brezzi-Rappaz-Raviart (BRR) approximation theorem in order to obtain existence and error estimates for the finite element approximations of the stable solutions. However, the $C^1$ regularity of $F$ requires the Hamiltonian $H$ to be of class $C^2$, which is rather restrictive from the point of view of optimal control. In order to generalize our approach to the more natural class of Hamiltonians with $C^{1,1}$ regularity, we generalize the BRR theorem to the case of maps $F$ that are Lipschitz continuous and satisfy a metric regularity assumption. This last assumption, coming from variational analysis, replaces the invertibility of $dF[\bar{x}]$. This generalization allows us to extend our error estimates for the finite element approximations of the stable solutions to the MFG system to the case of Hamiltonians of class $C^{1,1}$.

Semi-Lagrangian approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

• Luciano Marzufero (Libera Università di Bolzano)
• Adriano Festa (Politecnico di Torino)
• Fabio Camilli (Università "G. d'Annunzio" Chieti-Pescara)

Room: Aula Dini The aim of this work is to investigate semi-Lagrangian approximation schemes on unstructured grids for viscous transport and conservative equations with measurable coefficients that satisfy a one-sided Lipschitz condition. To establish the convergence of the schemes, we exploit the characterization of the solution for these equations expressed in terms of measurable time-dependent viscosity solution and, respectively, duality solution. We supplement our theoretical analysis with various numerical examples to illustrate the features of the schemes.

Mean convexity and the truncated Laplacian

• Martino Bardi (Università degli studi di Padova)

Room: Aula Dini We propose a notion of mean convexity of a function in R^n with respect to all k-dimensional subspaces, k integer between 1 and n. We show its connection with the truncated Laplacian of order k of the function, i.e., the sum of the first k eigenvalues of the Hessian matrix. We consider the Hessian PDE that prescribes the k-truncated Laplacian, in viscosity sense. We study existence and uniqueness of solutions to the Dirichlet problem under suitable conditions on the data, in particular convexity properties of the domain. Here we exploit some results by Birindelli, Galise and Ishii. Our main result is about the inverse problem of the single-valuedness of the truncated Laplacian in viscosity sense. Its solution allows us to characterise k-mean convexity of continuous functions in terms of inequalities for the Hessian equation. Joint work with P. Mannucci