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

Montefalcone, Francescopaolo

Some relations among volume, intrinsic perimeter and one-dimensional restrictions of

B V

functions in Carnot groups

Montefalcone, Francescopaolo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 1, p. 79-128

Abstract

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.

MR 2165404 | Zbl 1150.49022

Classification: 49Q15, 46E35, 22E60

@article{ASNSP_2005_5_4_1_79_0,
     author = {Montefalcone, Francescopaolo},
     title = {Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {1},
     year = {2005},
     pages = {79-128},
     zbl = {1150.49022},
     mrnumber = {2165404},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_1_79_0}
}

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, Serie 5, Volume 4 (2005) no. 1, pp. 79-128. http://www.numdam.org/item/ASNSP_2005_5_4_1_79_0/

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