American Institute of Mathematical Sciences

February  2013, 6(1): 167-191. doi: 10.3934/dcdss.2013.6.167

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

 1 Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I-25133 Brescia, Italy 2 Dipartimento di Matematica “F. Casorati", Università di Pavia, Via Ferrata 1, I- 27100 Pavia, Italy

Received  September 2011 Revised  October 2011 Published  October 2012

The notion of BV solution to a rate-independent system was introduced in [8] to describe the vanishing viscosity limit (in the dissipation term) of doubly nonlinear evolution equations. Like energetic solutions [5] 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.
Citation: Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167
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