Evolution variational inequalities and multidimensional hysteresis operators
Pavel Krejčı́
Abstract
One may wonder why such a particular problem like the variational inequality 1.1 https://www.w3.org/1998/Math/MathML" display="block"> u ˙ ( t ) − x ˙ ( t ) , x ( t ) − x ˜ � ≥ 0 ∀ x ˜ ∈ Z , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332555/ddb532f6-ff62-48a6-bdc1-effe7fe2ef8d/content/eqn0240.tif"/> where Z is a convex closed subset of a Hilbert space X, u is a given X-valued function of t ϵ [0, T], x is the unknown function with values in Z and dot denotes the derivative with respect to t, should draw exceptional attention. As in many analogous cases, it has been extracted as a common feature of different physical models. Its variational character is typically interpreted as a special form of the maximal dissipation principle in evolution systems with convex constraints. It turns out that inequalities of the form (1.1) play (explicitly or implicitly) a central role in modeling nonequilibrium processes with rate-independent memory in mechanics of elastoplastic and thermoelastoplastic materials including metals, polymers or for instance bread dough, as well as in ferromagnetism, piezoelectricity or phase transitions (see e.g. [DL, LC, Al, LT, NH, BS, V, Be, KS1, KS2, KS3, KS4, AGM]). They also naturally arise in the analysis of fatigue and damage accumulation (see [BDK, BS]).