Existence and uniqueness results for general rate-independent hysteresis problems∗

Here, E and Ψ are assumed to be lower semicontinuous, convex in their second arguments and differentiable in their first arguments, and the symbols ∂v and ∂ both denote the subdifferential w.r.t. the second variable. In fact, E is the potential energy and Ψ the dissipation functional associated with a rate-independent process, possibly displaying a hysteretic behaviour. Roughly speaking, rate-independence means that the process is insensitive to changes in the time scales. Processes of this kind occur in several branches of applied mathematics, such as plasticity, phase transformations in elastic solids, dry friction on surfaces and many others (see e.g., [Mie05] and the references therein). They may arise as vanishing viscosity limits of systems with strongly separated time scales, whence their hysteretical behaviour. On the modeling level, rate-independence is achieved by assuming Ψ to be 1-positively homogeneous w.r.t. its second variable, i.e., Ψ(z, λv) = λΨ(z, v) for every λ ≥ 0 and (z, v) ∈ Z × Z. Thus, a solution to (1.1) remains a solution if the time is rescaled. In the last years, a new energetic approach to the modeling of these problems has been developed in [MT99, MTL02, MT04]. The latter work concerns a simplified version of (1.1), obtained by assuming that Ψ does not depend on the state z, i.e., DzΨ(z, v) = 0 for all z, v. This leads to a special case of the doubly nonlinear problems studied in [CV90, Col92] because of the additional rate independence. It is the purpose of this paper to generalize the results in [MT04], proving existence, approximation, and uniqueness for (1.1), which includes the state-dependent dissipation functional Ψ. From the very beginning, we will assume the map z 7→ E(t, z) to be convex: this is necessary to obtain ∗Supported by EU via HPRN-CT-2002-00284 Smart Systems †Weierstraß-Institut, Mohrenstraße 39, 10117 D–Berlin and Institut für Mathematik, HumboldtUniversität zu Berlin, Rudower Chaussee 25, D–12489 Berlin (Adlershof) ‡Dipartimento di Matematica, Università di Brescia, Via Valotti 9, 20133 Brescia, Italy