Curvature-dependent energies: the elastic case
The authors continue the analysis begun in [the authors, ``Curvature dependent energies: a geometric and analytical approach'', Proc. R. Soc. Edinb., Sect. A, Math. (to appear), \url{doi:10.1017/S0308210516000202}]. In the present paper the authors study the curvature functional for non-smooth Cartesian curves in codimension higher than one. They use the Gauss map \(\tau_u:I\to \mathbb{S}^N\) defined as \(\tau_u=\dot c_u/|\dot c_u|, \;\dot c_u=(1,\dot u^1,\dots \dot u^N)\), the total curvature \(TC(c)\) of a curve considered by Milnor (in the case of Cartesian curves we have \(TC(c_u)=\int_{c_u}k_{c_u}d\mathcal{H}^1=\int_I|\tau_u|dt\)). The authors define the \(p\)-curvature functional of smooth Cartesian curves as \(TCP_p(c_u)=\int |\dot c_u|^{1-p}|\dot \tau_u|^pdt, \;p>1\). They extend the definition to the non-smooth case of continuous functions \(u\) with relaxed energy and with no corner points. After introducing the energy functional in Section 2, they recall the definition of Gauss graph of Cartesian curves. Next they deal, in Section 3, with the energy functional \(\mathcal{E}_p^0\) on currents, following an approach by \textit{M. Giaquinta} et al. [Cartesian currents in the calculus of variations I. Cartesian currents. Berlin: Springer (1998; Zbl 0914.49001); Cartesian currents in the calculus of variations II. Variational integrals. Berlin: Springer (1998; Zbl 0914.49002)]. Then the authors present some structure properties of the class of currents that naturally arise in the relaxation process. Then they introduce a class of minimal currents associated to the considered relaxation problem. The following energy lower bound holds: the relaxed energy of \(u\) is greater than the energy of the corresponding minimal current with the underlying function \(u\). In Section 6, the authors outline some features concerning functions with finite relaxed energy. In particular, they prove that (in high codimenion) the set of corner points is always finite. The energy upper bound is obtained in Section 7, by means of a suitable approximation result. As a consequence, in Section 8, the authors prove that the Gauss map \(\tau_u\) of a function with finite relaxed energy is a BV-function with no Cantor part. Therefore, if \(u\) is continuous and with no corner points, then \(\tau_u\) is a Sobolev function. In the last section, the authors present a summary of the obtained results.
scientific article; zbMATH DE number 6695863
scientific article