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BV solutions of rate independent differential inclusions. (English) Zbl 1349.34243
For $$H$$ a real Hilbert space, $$0 \in Z \varsubsetneq H$$ closed, convex, and $$z_0 \in H$$, the stop operator $$S : W^{1,1}((0,T);H) \to W^{1,1}((0,T);H)$$ associates to $$u$$, the solution $$x=S(u)$$ to the following evolution differential inclusion: \begin{aligned} &x(t) \in Z \quad \forall t \in [0,T];\\ &u'(t) \in x'(t) + \partial I_Z(x(t)) \quad a.e.\;t\in [0,T],\\ &x(0) = z_0.\end{aligned} Some extensions of $$S$$ to the space of functions of bounded variation have been obtained. The aim of this paper is to recall and compare these BV solutions. A geometric characterization in terms of $$Z$$ of the cases when all these extensions are equivalent is provided in the finite dimensional case.

##### MSC:
 34G25 Evolution inclusions 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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