A Result About $C^2$-Rectifiability of One-Dimensional Rectifiable Sets. Application to a Class of One-Dimensional Integral Currents

Delladio, Silvano

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 237-252 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

Let $\gamma, \tau \colon [a, b] \rightarrow R^{k+1}$ be a couple of Lipschitz maps such that $\gamma' = \pm |\gamma'|\tau$ almost everywhere in $[a, b]$. Then $\gamma([a, b])$ is a $C^2$-rectifiable set, namely it may be covered by countably many curves of class $C^2$ embedded in $R^{k+1}$. As a conseguence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a $C^2$-rectifiable set.

Résumé

Siano $\gamma, \tau \colon [a, b] \rightarrow R^{k+1}$ due mappe Lipschitziane tali che $\gamma' = \pm |\gamma'|\tau$, quasi ovunque in $[a, b]$. Allora $\gamma([a, b])$ è un insieme $C^2$-rettificabile, ossia esso è incluso (eccetto per un insieme di misura nulla) in una unione numerabile di sottovarietà uno-dimensionali di $R^{k+1}$ di classe $C^2$. Di conseguenza, la proiezione del carrier rettificabile di un grafico di Gauss generalizzato uno-dimensionale è un insieme $C^2$- rettificabile.

Détail

Publié le : 2007-02-01

MR 2310966 | Zbl 1178.53003

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