Fausto Ferrari

Harnack inequality for fractional sub-Laplacians in Carnot groups

In this paper we prove an invariant Harnack inequality on Carnot–Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an “abstract” formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups.

Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems

Let be a divergence form operator with Lipschitz continuous coefficients in a domain, and let be a continuous weak solution of in. In this paper, we show that if satisfies a suitable differential inequality, then is a subsolution of away from its zero set. We apply this result to prove regularity of Lipschitz free boundaries in two-phase problems.

On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.

The Geometric Traveling Salesman Problem in the Heisenberg Group

In the Heisenberg group H (endowed with its Carnot-Carathéodory structure), we prove that a compact set E ⊂ H which satisfies an analog of Peter Jones’ geometric lemma is contained in a rectifiable curve. This quantitative condition is given in terms of Heisenberg β numbers which measure how well the set E is approximated by Heisenberg straight lines.

Two-phase problems with distributed sources: regularity of the free boundary

We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are

Regularity of the Solutions for Parabolic Two-Phase Free Boundary Problems

C^{1,\gamma}

Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients

. In particular, viscosity solutions are indeed classical.

A Local Doubling Formula for the Harmonic Measure Associated with Subelliptic Operators and Applications

In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary.

Hessian inequalities and the fractional Laplacian

In this paper we prove that flat free boundaries of the solutions of elliptic two-phase problems associated with a class of fully nonlinear operators are C1,γ . In [11] the C1,γ regularity of Lipschitz free boundaries of two-phase problems was proved for a class of homogeneous fully nonlinear elliptic operators F(D2u(x), x), containing convex (concave) operators, with Holder dependence on x. Here we consider the same class of operators. More precisely, we prove the regularity of flat free boundaries of the solutions of the following two-phase problems:  F(D2u(x), x) = 0 in Ω(u) = {x ∈ Ω ⊂ R : u > 0}, F (D2u(x), x) = 0 in Ω(u) = {x ∈ Ω ⊂ R : u 6 0}, u = 0 on Fu, u+ν = G(u − ν ) on Fu, (1)

An extension problem for the fractional derivative defined by Marchaud

Abstract Let X={ X1, …, Xp} be a family of smooth vector fields in R n, and let Ω⊂R n be a connected open subset with sufficiently regular boundary ∂Ω. Assume X satisfies the so-called Hörmander rank condition, i.e., suppose the rank of the Lie algebra generated by X equals n at any point of a neighborhood of . In this paper we prove a doubling property, with respect to a suitable family of neighborhoods of a boundary point, of the harmonic measure associated with the subelliptic operator . As a consequence, we derive a boundary Harnack principle for positive weak solutions of . †To Professor Jean-Michel Bony on his 60th birthday