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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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