Curvature-dependent energies: The elastic case

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.