Curvature-dependent Energies

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 of continuous functions u on an interval, the relaxed energy is finite if and only if the curve 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.