Federica Dragoni

• PhD in Mathematics
• Professor (Full) at Cardiff University

We show that the square of Carnot–Carathéodory distance from the origin, in step 2 Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We first give a proof in the case of ideal Carnot groups, based on the simple group structure as well as estimates for the Euclidean semiconcavity. Our...

We give an overview on semiconcavity, starting from the standard notion, up to more recent generalizations in a different geometrical context, such as Carnot groups; focusing in particular on the viscosity characterization by bounds for second derivatives. We then apply these theories to show some recent results obtained by the author, in collabora...

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonloca...

We show that the square of Carnot-Carath\'eodory distance from the origin, in step 2 Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We first give a proof in the case of ideal Carnot groups, based on the simple group structure as well as estimates for the Euclidean semiconcavity. Ou...

We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge Θ for which we introduce a suitable notion of Legendre transform Θ_0. We compute the action of the relevant nonlinear PDEs on "radial" f...

The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situat...

The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situat...

This paper is on Γ-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coerciv...

This paper is on Γ-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coerciv...

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 Hörmander's case. We develop some new notions and investigate their mutual relations and properties.

We prove results on existence, uniqueness and regularity of solutions for a class of systems of subelliptic PDEs arising from Mean Field Game systems with Hörmander diffusion. These results are applied to the feedback synthesis Mean Field Game solutions and the Nash equilibria of a large class of N-player differential games.

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.

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.

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.

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...

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...

We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Caratheodory metric spaces. We define a suitable subdifferential and show that a continuous function is X-convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce...

We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in Carnot- Carathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in...

Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous wi...

We develop an effective strategy for proving strong ergodicity of (nonsymmetric) Markov semigroups associated to H\"ormander type generators when the underlying configuration space is infinite dimensional.

We introduce a notion of convexity in the geometry of vector fields and we prove a PDE-characterization for such notion and related properties.

In ℝ n equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization for when the distance from the origin in an arbitrary Carnot-Carathéodory space is a viscosity infinite harmonic function at a point outside the origin. We show t...

We study the phenomenon of evolution by horizontal mean curvature flow in sub-Riemannian geometries. We use a stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for...

We investigate the behavior, as ε → 0+, of ε log w ε (t, x) where w ε are solutions of a suitable family of subelliptic heat equations. Using the Large Deviation Principle, we show that the limiting behavior is described by the metric inf-convolution w.r.t. the associated Carnot-Carathéodory distance.

(Communicated by Giuseppe Buttazzo) Abstract. In this paper we study a metric Hopf-Lax formula looking in particular at the Carnot-Carathéodory case. We generalize many properties of the classical euclidean Hopf-Lax formula and we use it in order to get existence results for Hamilton-Jacobi-Cauchy problems satisfying a suitable Hörmander condition....

An inverse problem of photon transport in a dusty medium with slab symmetry is studied. The problem consists in finding the unknown densities of two different kinds of dust from measurements of radiation intensities at two different frequencies. Under suitable assumptions, the problem is shown to have a unique solution. Some numerical experiments a...