zbMATH — the first resource for mathematics

Evolution variational inequalities and multidimensional hysteresis operators. (English) Zbl 0949.47053
Drábek, Pavel (ed.) et al., Nonlinear differential equations. Proceedings of talks given at the seminar in differential equations, Chvalatice, Czech Republic, June 29-July 3, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 404, 47-110 (1999).
Analytical properties of hysteresis operators are discussed. A fundamental example for a hysteresis operator is the stop operator \(S\): Let \(Z\) be a closed convex set of a Hilbert space \(X\) with \(0\), \(x_0\in Z\). The operator \(S\) assignes to \((x_0,u)\) with \(u\in W^{1,1}(0,T;X)\) the function \(x\in W^{1,1}(0, T;Z)\) satisfying \[ \langle\dot u(t)-\dot x(t), x(t)-\widetilde x\rangle\geq 0 \] for all \(\widetilde x\in Z\). Hysteresis operators are used in the modeling of time dependent processes with rate independent memory. An important example for such a process is the deformation of an elastic-plastic meterial.
The article is mainly devoted to the investigation of the influence of the geometry of \(Z\) on the analytical properties of the hysteresis operator. The article has an overview character, since most results are taken from the books [M. A. Krasnosel’skij and A. V. Pokrovskij: “Systems with hysteresis”. Berlin (1989; Zbl 0665.47038), P. Krejčí: “Hysteresis, convexity and dissipation in hyperbolic equations”. Tokyo: Gakkōtosho (1996)], but it also contains new results. The results stated in the article are completely proved.
For the entire collection see [Zbl 0919.00053].

47J20 Variational and other types of inequalities involving nonlinear operators (general)
47J40 Equations with nonlinear hysteresis operators
35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators