Existence of Solutions for Degenerate Sweeping Processes
cf. e.g. , and also [11]. In this formulation, every E(x) ⊂ IR is a closed convex set (sometimes called the ”rigidity set”), t ≥ 0 denotes the process time, x ∈ Ω ⊂ IR is the material point, q ∈ IR an internal variable (like plastic strain or hardening), and W denotes a (modified) stored energy function. For our purpose of motivation, the main thing to note is that for every fixed x, the inclusion (2) is of type (1).

title: ON THE DISCRETIZATION OF DEGENERATE SWEEPING PROCESSES
abstract: We prove existence theorems for evolution equations of the form −u(t) ∈ ∂δC(t)(Au(t)) with some maximal monotone and strongly monotone operator A : D(A)→ 2 . 1 – Introduction and main results We study the evolution problem (1) −u(t) ∈ ∂δC(t)(Au(t)) a.e. in [0, T ], u(0) = u0 ∈ D(A) , where A is a maximal monotone and strongly monotone operator in the real Hilbert space H, and t 7→ C(t) is a Hausdorff-continuous multifunction with closed convex values. Equations of this form arise from problems of the type −x(t) ∈ ∂δC(g(t, x(t))), which play an important rôle in elasticity theory, cf. [8, 3, 4] for more information. To solve (1) means that we have to find u ∈W 1,1([0, T ];H) and v ∈ L2([0, T ];H) such that u(0) = u0, u(t)∈D(A) a.e., v(t)∈Au(t)∩C(t) a.e. and −u′(t)∈∂δC(t)(v(t)) a.e. in [0, T ] . Our general assumptions are (H1) A:D(A)→2H\{∅} is a maximal monotone operator (abbreviated mmop) such that A = ∂ψ for some lsc, convex and proper ψ : H → R ∪ {∞}, and there exists a β > 0 such that (2) 〈Ax−Ay, x− y〉 ≥ β |x− y| for x, y ∈ D(A) .  December 31, 1996;  April 4, 1997.  (1991): 34A60, 35K22.

title: Yosida–Moreau Regularization of Sweeping Processes with Unbounded Variation
abstract: Lett↦C(t) be a Hausdorff-continuous multifunction with closed convex values in a Hilbert spaceHsuch thatC(t) has nonempty interior for allt. We show that the Yosida–Moreau regularizations of the sweeping process with moving setC(t), i.e., the solutions of[formula]are strongly pointwisely convergent asλ→0+to the solution of the corresponding sweeping process, formally written as[formula]

title: Unbounded Solutions in Non-Smooth Dynamical Systems at Resonance
abstract: We consider periodically forced non-smooth dynamical systems at resonance, described by differential inclusions, and we show that analogously to the case of ODEs all solutions are unbounded in the (x)-phase plane, if the first Fourier coefficient of the forcing is large compared to a certain quantity related to the nonlinearity.

title: On the Uniqueness and Asymptotic Behaviour of Plasma Corners

title: On parabolic quasi-variational inequalities and state-dependent sweeping processes
abstract: (1.2) C(t, u) ⊂ H is nonempty, closed, and convex for t ∈ [0, T ], u ∈ H. In (1.1), NC(t,u)(x) denotes the normal cone to C(t, u) at x ∈ C(t, u), cf. Section 2 below. We will treat the case of (t, u) 7→ C(t, u) being Lipschitz continuous w.r. to the Hausdorff distance dH with constants L1, L2 ≥ 0, i.e., we require (1.3) dH(C(t, u), C(s, v)) ≤ L1|t− s|+ L2|u− v|, t, s ∈ [0, T ], u, v ∈ H. Note that a solution of (1.1) in particular has to satisfy u(t) ∈ C(t, u(t)) for t ∈ [0, T ].

title: Bilateral semidirect products of transformation semigroups