Calculus of Variations and Geometric Measure Theory

E. Acerbi - D. Mucci

Curvature-dependent energies: a geometric and analytical approach

created by mucci on 24 Nov 2015
modified on 24 Apr 2017


Published Paper

Inserted: 24 nov 2015
Last Updated: 24 apr 2017

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Year: 2017

Published online: 27 February 2017


We consider the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs $c_u$ of continuous functions $u$ on an interval, we show that the relaxed energy is finite if and only if the curve $c_u$ has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of $u$. We treat the wider class of graphs of one-dimensional BV-functions, and we prove that the relaxed energy is given by the sum of length and total curvature of the new curve obtained by closing with vertical segments the holes in $c_u$ generated by jumps of $u$.