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

Riccarda Rossi; Giuseppe Savaré

doi:10.3934/dcdss.2013.6.167

Article Contents

2013, Volume 6, Issue 1: 167-191. Doi: 10.3934/dcdss.2013.6.167

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

Riccarda Rossi^1, and
Giuseppe Savaré^2,

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

Received: August 31, 2011

Revised: September 30, 2011

Published: January 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C55, 47J20, 49J40; Secondary: 74N30.

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References

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