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On a class of scalar variational inequalities with measure data. (English) Zbl 1177.74308
Summary: We consider a class of variational inequalities defining the so-called hysteresis play operator. We propose a new approach to discontinuous \(BV\)-solutions based on measure theoretical arguments, which enable us to infer the existence of solutions as a simple consequence of the classical theory. In this way, we generalize a recent result where only continuous \(BV\)-solutions were studied. We also provide a representation formula which allows to deduce the continuity of the play operator from general theorems on hysteresis operators.

MSC:
74N30 Problems involving hysteresis in solids
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
34C55 Hysteresis for ordinary differential equations
26A45 Functions of bounded variation, generalizations
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