Calculus of Variations and Geometric Measure Theory

S. Delladio

A result about $C^2$-rectifiability of one-dimensional rectifiable sets. Application to a class of one-dimensional integral currents

created by delladio on 25 Jul 2006


Accepted Paper

Inserted: 25 jul 2006

Journal: Boll. Un. Matem. Italiana
Year: 2006


Let $g, t$ be a couple of Lipschitz $*R*^{k+1}$-valued maps defined in an interval $[a,b]$ and such that $Dg=\pm\vert Dg\vert t$ almost everywhere in $[a,b]$. Then $g([a,b])$ 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 consequence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a $C^2$-rectifiable set.

Keywords: Rectifiable sets, Geometric measure theory, non-homogeneous blow-ups, Whitney extension theorem