BV‐extension of rate independent operators

Abstract

Key words Rate independent operators, hysteresis, extending rate independent operators, functions of bounded variation MSC (2000) 47H30, 47H99, 74N30Rate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones are those better suited for the study of PDE's with hysteresis. We prove that a rate independent operator R : Lip(0, T ) → BV (0, T ) ∩ C(0, T ) which is locally monotone and continuous with respect to the strict topology of BV admits a unique continuous extension R : BV (0, T ) → BV (0, T ). This general result applies to several concrete hysteresis operators. For many of these operators the existence of a continuous extension was previously known at most to the space BV (0, T ) ∩ C(0, T ).

The play operator on the rectifiable curves in a Hilbert space

“…Hence, we prove directly that P( u)• u is the solution of (3) We found formula (7) in [7] where we look for sufficient conditions in order to extend a general scalar rate-independent operator R :…”

“…such that R( [7] and [8] we prove that given a rate-independent operator that is continuous with respect to the strict metric, then it can be continuously extended to BV ([0, T ]; R) if and only if it is locally isotone, i.e. if (8) is not needed and the existence of the continuous extension R : …”

“…In [10] a definition of solution for u with bounded variation is given and it is very similar to the integral condition (3): the integral has to be understood in the Young sense, and the integrand map has to be replaced with its right continuous representative. However, it is possible to prove that in general for discontinuous u ∈ BV ([0, T ]; H), the unique solution found in [10] is not given by formula (7). The relationship between P( u)• u and the solution of [10] is investigated in [9].…”

BV-norm continuity of sweeping processes driven by a set with constant shape

“…In general P is not BV-strict continuous on the whole BV r ([0, T ] ; H), it was proved in [39] that the continuity in the strict topology holds if and only if Z = {x ∈ H : −α ≤ f, x ≤ β} for some f ∈ H {0} and α, β ∈ [0, ∞]. In the one dimensional case it turns out P is always BV-strict continuous on BV r ([0, T ] ; R) (see also [47,9,37,38]). …”

Sobolev and strict continuity of general hysteresis operators

“…This is in particular important in order to investigate possible extensions to the space of functions of bounded variation, indeed in [6,7] it is proved that every R : W 1,1 (I ) −→ W 1,1 (I ), which is continuous with respect to the strict metric, can be uniquely extended…”