On locally isotone rate independent operators.

*(English)*Zbl 1152.47059The extension of rate-independent operators (for instance, hysteresis operators) to domains of non continuous functions is an important issue for applications as well as a challenging problem from an analytical point of view.

In this interesting paper, the author, using a previous sufficiency result from his previous work, gives a complete characterization of those rate independent operators which admit a continuous extension to the space of functions of bounded variation. In particular, the result may be expressed as follows. Let us be given of a set of continuous time-dependent functions of bounded variation, containing the set of Lipschitz functions. Then a rate-independent operator on it, which is continuous with respect to the strict topology of \(BV\), and which maps Lipschitz functions to continuous functions, may be continuously extended to \(BV\) if and only if it is locally isotone.

Here, isotone means that the operator locally maintains the monotonicity (but not necessary the sign of it). Also, a representation of such extensions is given.

In this interesting paper, the author, using a previous sufficiency result from his previous work, gives a complete characterization of those rate independent operators which admit a continuous extension to the space of functions of bounded variation. In particular, the result may be expressed as follows. Let us be given of a set of continuous time-dependent functions of bounded variation, containing the set of Lipschitz functions. Then a rate-independent operator on it, which is continuous with respect to the strict topology of \(BV\), and which maps Lipschitz functions to continuous functions, may be continuously extended to \(BV\) if and only if it is locally isotone.

Here, isotone means that the operator locally maintains the monotonicity (but not necessary the sign of it). Also, a representation of such extensions is given.

Reviewer: Fabio Bagagiolo (Trento)

##### MSC:

47J40 | Equations with nonlinear hysteresis operators |

47J99 | Equations and inequalities involving nonlinear operators |

26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |

##### Keywords:

rate independent operators; hysteresis; extending rate independent operators; functions of bounded variation
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\textit{V. Recupero}, Appl. Math. Lett. 20, No. 11, 1156--1160 (2007; Zbl 1152.47059)

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##### References:

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[2] | Visintin, A., () |

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[5] | V. Recupero, \(B V\)-extension of rate independent operators, Math. Nachr. (in press) · Zbl 1168.47049 |

[6] | Logemann, H.; Mawby, A.D., Extending hysteresis operators to spaces of piecewise continuous functions, J. math. anal. appl., 282, 107-127, (2003) · Zbl 1042.47049 |

[7] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variation and free discontinuity problems, (2000), Clarendon Press Oxford · Zbl 0957.49001 |

[8] | Federer, H., Geometric measure theory, (1969), Springer-Verlag Berlin, Heidelberg · Zbl 0176.00801 |

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