Calculus of Variations and Geometric Measure Theory

E. Acerbi - D. Mucci

Curvature-dependent energies: the elastic case

created by mucci on 04 Apr 2016
modified on 24 Apr 2017


Published Paper

Inserted: 4 apr 2016
Last Updated: 24 apr 2017

Journal: Nonlinear Analysis
Volume: 153
Pages: 7--34
Year: 2017


We continue our analysis on functionals depending on the curvature of graphs of curves in high codimension Euclidean space. We deal with the ``elastic" case, corresponding to a superlinear dependence on the pointwise curvature. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. Different phenomena w.r.t. the "plastic" case, i.e. to the relaxation of the total curvature functional, 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. The wider class of graphs of one-dimensional BV-functions is treated.

Keywords: Curvature, Cartesian currents, Image restoration