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Some relations among volume, intrinsic perimeter and one-dimensional restrictions of B V functions in Carnot groups

Francescopaolo Montefalcone — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let 𝔾 be a k -step Carnot group. The first aim of this paper is to show an interplay between volume and 𝔾 -perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for 𝔾 -regular submanifolds of codimension one. We then give some applications of this result: slicing of B V 𝔾 functions, integral geometric formulae for volume and 𝔾 -perimeter and, making use of a suitable notion of convexity, called, we state a Cauchy type formula for 𝔾 -convex sets. Finally, in the last...

Sets of finite perimeter associated with vector fields and polyhedral approximation

Francescopaolo Montefalcone — 2003

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let X = X 1 , ⋯ , X m be a family of bounded Lipschitz continuous vector fields on R n . In this paper we prove that if E is a set of finite X -perimeter then his X -perimeter is the limit of the X -perimeters of a sequence of euclidean polyhedra approximating E in L 1 -norm. This extends to Carnot-Carathéodory geometry a classical theorem of E. De Giorgi.

Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups

Francescopaolo Montefalcone — 2016

Analysis and Geometry in Metric Spaces

In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition.

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