In the nonconvex case, solutions of rate-independent systems may develop jumps as afunction of time. To model such jumps, we adopt the philosophy that rate-independenceshould be considered as limit of systems with smaller and smaller viscosity. For thefinite-dimensional case we study the vanishing-viscosity limit of doubly nonlinearequations given in terms of a differentiable energy functional and a dissipation potentialthat is a viscous regularization of a given rate-independent dissipation potential. Theresulting definition of “BV solutions” involves, in a nontrivial way, both therate-independent and the viscous dissipation potential, which play crucial roles in thedescription of the associated jump trajectories. We shall prove general convergenceresults for the time-continuous and for the time-discretized viscous approximations andestablish various properties of the limiting BV solutions. In particular, we shall providea careful description of the jumps and compare the new notion of solutions with therelated concepts of energetic and local solutions to rate-independent systems.
