Schedule

Abstract: The long-time average behaviour of the value function in the classical calculus of variations is known to be connected with the existence of solutions of the so-called critical equation, that is, a stationary Hamilton-Jacobi which includes a sort of nonlinear eigenvalue called the critical constant or effective Hamiltonian. Here, we will address similar issues for sub-Riemannian systems, that is, control systems associated with a family of vector fields which satisfies the Lie Algebra rank condition on the whole Euclidean space. In particular, we will construct the critical constant by studying the convergence of the time-averaged value function as time horizon goes to infinity. We will also obtain a variational representation of such a constant by an adapted notion of closed measures, which is a class of measures introduced by A. Fathi and A. Siconolfi in [Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 2(155):363–388, 2004].

Abstract: The aim of this talk is to clarify the intimate relations between Lyapunov functions and chain recurrent sets. The study of this subject comes from a seminal paper by Conley and has had recent advances by the so-called “Weak KAM Theory for homeomorphisms” by Fathi and Pageault. The talk intends to give an explanation of the state of the art, discuss connections with the stationary Hamilton-Jacobi equation and present recent improvements. Joint works with A. Florio and J. Wiseman.

Abstract: The talk is built upon the following general question: does convexity reinforce the properties of the gradient trajectories of a function? Concerning this topic, I will recall a main positive result: bounded gradient trajectories have finite length. I will also discuss some negative aspects: convexity is irrelevant for the Lojasiewicz inequality (or more generally, for the Kurdyka-Lojasiewicz inequality), which is false in general. Moreover, the gradient trajectory of Thom may fail both around minima and at infinity. This is a joint work with Aris Daniilidis (Vienna) and Mounir Haddou (Rennes).

Abstract: We show a connection between global unconstrained optimization of a continuous function f and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution v of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then v is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of v. Such trajectories are proved to converge to the set of minima of f, using tools in control theory and occupational measures. This is joint work with Hicham Kouhkouh.

Abstract: Finite mixture models, given by a convex combination of probability density functions, are a powerful tool for statistical modeling of data, with applications to pattern recognition, computer vision, signal and image analysis, machine learning, etc. Given a finite data set, the corresponding mixture model can be computed by means of the Expectation-Maximization (EM) algorithm, a classical optimization technique which incrementally converges to a maximum likelihood estimate of the model's parameters. In this talk, I will present a multi-population Mean Field Games systems which can be intepreted as an infinite-dimensional version of the EM algorithm. I will discuss the theoretical aspects of the method and the application to some problems in cluster analysis.

Abstract: I will present some results on the long time behavior of fractional mean curvature flow. In particular I will discuss the evolution of entire Lipschitz graphs and show stability of hyperplanes and mean convex cones. Moreover I will consider the volume constrained fractional mean curvature flow of a nearly spherical set, showing long time existence and exponential convergence to a ball. The result applies also to convex initial data, under the assumption of global existence.

Abstract: Let $H : \mathbb{T}^N \times \mathbb{R}^N\to \mathbb{R}$ be a continuous, coercive Hamiltonian that is convex in the second variable. We will present some existence / uniqueness results for viscosity solutions of equations of the form

$$\lambda \alpha(x) u(x) + H(x, D_x u) = 0$$

Where $\lambda>0$, is a constant, $\alpha : \mathbb{T}^N \to. [0,+\infty)$ is continuous and $u : \mathbb{T}^N \to \mathbb{R}$ is the unknown function.

Such results involve conditions on the non—vanishing set of $\alpha$ that highlights the fundamental role of Mather measures. When this condition is satisfied, we will also present the convergence of solutions of those discounted equations as the discount factor $\lambda \to 0$.

We will also describe more general settings where such results hold. This is based on a joint work with Q. Chen, A. Fathi and J. Zhang.

Abstract: We present recent results about radial solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. In particular, we show that, for the existence of sign-changing solutions, a new critical exponent appears. Furthermore, we analyze the new concentration phenomena occurring as the exponents approach the critical values.

Abstract: We provide a Lax–Oleinik-type representation formula for solutions of time–dependent Hamilton–Jacobi equations, posed on a network with a rather general geometry, under standard assumptions on the Hamiltonians. It depends on a given initial datum at t=0 and a flux limiter at the vertices, which both have to be assigned in order the problem to be uniquely solved. Previous results in the same direction are solely in the frame of junction, namely network with a single vertex. The results presented are otained in a joint work with A. Siconolfi.

Abstract: I discuss about the vanishing discount problem for Hamilton-Jacobi equations. After a brief review of the developments in the discount problem for Hamilton-Jacobi equations, I present an example of monotone systems of Hamilton-Jacobi equations, for which the full convergence of the solutions fails as the discount factor goes to zero.