Ergodic Mean Field Games with H\"ormander diffusions

We prove existence of solutions for a class of systems of subelliptic PDEs arising from Mean Field Game systems with H\"ormander diffusion. These results are motivated by the feedback synthesis Mean Field Game solutions and the Nash equilibria of a large class of $N$-player differential games

Starshapedeness for fully-nonlinear equations in Carnot groups

In this paper we establish the starshapedness of the level sets of the capacitary potential of a large class of fully-nonlinear equations for condensers in Carnot groups, once a natural notion of starshapedness has been introduced. Our main result is Theorem 1.2 below.Comment: Accepted for publication in the Journal of the London Mathematical Societ

Stochastic homogenization for functionals with anisotropic rescaling and non-coercive Hamilton-Jacobi equations

We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the $\varepsilon$-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem

Starshaped and convex sets in Carnot groups and in the geometries of vector fields

The paper gives an overview on convex sets and starshaped sets in Carnot groups and in the more general case of geometries of vector fields, including in particular the H¨ormander’s case. We develop some new notions and investigate their mutual relations and propertie

Starshapedeness for fully non-linear equations in Carnot groups

In this paper, we establish the starshapedness of the level sets of the capacitary potential of a large class of fully nonlinear equations for condensers in Carnot groups

Oxidative Stress and Immune System in Vitiligo and Thyroid Diseases.

Vitiligo is an acquired dermatological disease frequently associated with autoimmune thyroid disorders. Several theories have been proposed so far to unravel the complex vitiligo pathogenesis. Currently, the autocytotoxic and the autoimmune theories are the most accredited hypothesis, since they are sustained by several important clinical and experimental evidences. A growing body of evidences shows that autoimmunity and oxidative stress strictly interact to finally determine melanocyte loss. In this scenario, associated thyroid autoimmunity might play an active and important role in triggering and maintaining the depigmentation process of vitiligo