Bruno Franchi

On the structure of finite perimeter sets in step 2 Carnot groups

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure.

Weighted sobolev-poincaré inequalities for grushin type operators

(1994). Weighted sobolev-poincare inequalities for grushin type operators. Communications in Partial Differential Equations: Vol. 19, No. 3-4, pp. 523-604.

A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type

The purpose of this note is to study the relationship between the validity of L1 versions of Poincare’s inequality and the existence of representation formulas for functions as (fractional) integral transforms of first-order vector fields. The simplest example of a representation formula of the type we have in mind is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space R:

Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations

In this paper we prove a Sobolev-Poincare inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Holder regularity of the weak solutions follows in a standard way. Let y = Z,7 ai(aij9 a ) be a second-order degenerate elliptic operator in divergence form with measurable coefficients. In this paper we shall obtain pointwise estimates for the weak solutions of Su = 0 (H61der continuity of the weak solutions and Harnack inequality for nonnegative solutions). Let us recall that the original results for elliptic operators were obtained by De Giorgi, Nash, and Moser. An extensive bibliography about the degenerate case can be found in [FLI, FL2, FS]. To introduce the results of the present paper, let us recall some recent results. In [FL1, FL2] a suitable metric d is associated with the differential operator Y in such a way that we obtain a new geometry which is natural for the degenerate operator as the Euclidean geometry is natural for the Laplace operator (or, more precisely, as a suitable Riemannian geometry is natural for a secondorder elliptic operator). In the smooth case, this idea is contained in many papers: we refer to [FP, NSW]. The basic results in [FL 1, FL2] are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the Hormander hypoellipticity condition for sum-of-squares operators. We note that the same idea is used in [NSW, S, J, V] to obtain pointwise estimates for sum-of-squares operators. On the other hand, a different class of degenerate elliptic operators is considered in [FKS]: instead of a geometrical degeneracy, a measure degeneracy is allowed. A typical example of this class is given by Yu = div(wo(x)Vu), where cl is a weight function belonging to the A2-class of Muckenhoupt. Unified results for a class containing both the operators in [FL 1] and in [FKS] have Received by the editors August 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E35, 35J70. Partially supported by G.N.A.F.A. of C.N.R. and M.U.R.S.T., Italy. ( 1991 American Mathematical Society 0002-9947/91

Weighted Poincaré Inequalities for Hörmander Vector Fields and local regularity for a class of degenerate elliptic equations

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Harnack inequality for fractional sub-Laplacians in Carnot groups

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The Geometric Traveling Salesman Problem in the Heisenberg Group

In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnacks inequality) for a class of degenerate elliptic equations with measurable coefficients.

div-curl Type Theorem, H-Convergence, and Stokes Formula in the Heisenberg Group

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.

Alzheimer's disease: a mathematical model for onset and progression

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.

Irregular Solutions of Linear Degenerate Elliptic Equations

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumins complex on contact manifolds.