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

Francescopaolo Montefalcone[1]

  • [1] Dipartimento di Matematica Università degli Studi di Bologna Piazza di P. ta S. Donato, 5 40126 Bologna, Italia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 1, page 79-128
  • ISSN: 0391-173X

Abstract

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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 𝔾 -convexity, we state a Cauchy type formula for 𝔾 -convex sets. Finally, in the last section we prove a sub-riemannian Santaló formula showing some related applications. In particular we find two lower bounds for the first eigenvalue of the Dirichlet problem for the Carnot sub-laplacian Δ 𝔾 on smooth domains.

How to cite

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Montefalcone, Francescopaolo. "Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.1 (2005): 79-128. <http://eudml.org/doc/84557>.

@article{Montefalcone2005,
abstract = {Let $\mathbb \{G\}$ be a $k$-step Carnot group. The first aim of this paper is to show an interplay between volume and $\mathbb \{G\}$-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for $\mathbb \{G\}$-regular submanifolds of codimension one. We then give some applications of this result: slicing of $BV_\{\mathbb \{G\}\}$ functions, integral geometric formulae for volume and $\mathbb \{G\}$-perimeter and, making use of a suitable notion of convexity, called$\mathbb \{G\}$-convexity, we state a Cauchy type formula for $\mathbb \{G\}$-convex sets. Finally, in the last section we prove a sub-riemannian Santaló formula showing some related applications. In particular we find two lower bounds for the first eigenvalue of the Dirichlet problem for the Carnot sub-laplacian $\Delta _\{\mathbb \{G\}\}$ on smooth domains.},
affiliation = {Dipartimento di Matematica Università degli Studi di Bologna Piazza di P. ta S. Donato, 5 40126 Bologna, Italia},
author = {Montefalcone, Francescopaolo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {measures on Carnot groups; -convexity; Fubini-like theorems},
language = {eng},
number = {1},
pages = {79-128},
publisher = {Scuola Normale Superiore, Pisa},
title = {Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups},
url = {http://eudml.org/doc/84557},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Montefalcone, Francescopaolo
TI - Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 1
SP - 79
EP - 128
AB - Let $\mathbb {G}$ be a $k$-step Carnot group. The first aim of this paper is to show an interplay between volume and $\mathbb {G}$-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for $\mathbb {G}$-regular submanifolds of codimension one. We then give some applications of this result: slicing of $BV_{\mathbb {G}}$ functions, integral geometric formulae for volume and $\mathbb {G}$-perimeter and, making use of a suitable notion of convexity, called$\mathbb {G}$-convexity, we state a Cauchy type formula for $\mathbb {G}$-convex sets. Finally, in the last section we prove a sub-riemannian Santaló formula showing some related applications. In particular we find two lower bounds for the first eigenvalue of the Dirichlet problem for the Carnot sub-laplacian $\Delta _{\mathbb {G}}$ on smooth domains.
LA - eng
KW - measures on Carnot groups; -convexity; Fubini-like theorems
UR - http://eudml.org/doc/84557
ER -

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