A characterization of energetic and BV solutions to one-dimensional rate-independent systems

The notion of BV solution to a rate-independent system was intro- duced in order to describe the vanishing viscosity limit (in the dissipation term) of doubly nonlinear evolution equations. Like energetic solutions in the case of convex energies, BV solutions provide a careful description of rate- independent evolution driven by nonconvex energies, and in particular of the energetic behavior of the system at jumps. In this paper we study both notions in the one-dimensional setting and we obtain a full characterization of BV and energetic solutions for a broad family of energy functionals. In the case of monotone loadings we provide a simple and explicit characterization of such solutions, which allows for a direct comparison of the two concepts.