Calculus of Variations and Geometric Measure Theory

E. Acerbi - D. Mucci

Curvature-dependent energies

created by mucci on 24 Apr 2017


Published Paper

Inserted: 24 apr 2017
Last Updated: 24 apr 2017

Journal: Milan J. Math.
Year: 2017
Doi: 10.1007/s00032-017-0265-x

Lecture given at the Seminario Matematico e Fisico di Milano by Emilio Acerbi on October 19, 2015.


We report our recent results on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs $c_u$ of continuous functions $u$ on an interval, 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 also deal with the ''elastic'' case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the ''plastic'' case are observed. A $p$-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the $p$-curvature. We treat the wider class of graphs of one-dimensional BV-functions.