Sets of finite perimeter associated with vector fields and polyhedral approximation

Montefalcone, Francescopaolo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 279-295 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

Let $X = X_{1}, \cdots, X_{m}$ be a family of bounded Lipschitz continuous vector fields on $\mathbb{R}^{n}$. 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.

Résumé

Sia data in $\mathbb{R}^{n}$ una m-upla $X = X_{1}, \cdots, X_{m}$ di campi vettoriali lipschitziani e limitati. In questo lavoro dimostriamo che se $E$ è un insieme di $X$-perimetro finito allora l’$X$-perimetro di $E$ è il limite degli $X$-perimetri di una successione di poliedrali euclidee approssimanti $E$ in norma $L^{1}$. Questo risultato estende alle geometrie di tipo Carnot-Carathéodory un classico teorema di E. De Giorgi.

Détail

Publié le : 2003-12-01

MR 2104216 | Zbl 1072.49031

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