17783663
Existence of Solutions for Degenerate Sweeping Processes
cf. e.g. [4], 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).
18597173
ON THE DISCRETIZATION OF DEGENERATE SWEEPING PROCESSES
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) . Received : December 31, 1996; Revised : April 4, 1997. Mathematical Subject Classification (1991): 34A60, 35K22.
122068311
Yosida–Moreau Regularization of Sweeping Processes with Unbounded Variation
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]
120017422
Unbounded Solutions in Non-Smooth Dynamical Systems at Resonance
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.
126071351
On the Uniqueness and Asymptotic Behaviour of Plasma Corners
14672702
On parabolic quasi-variational inequalities and state-dependent sweeping processes
(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 ].
121961938
Bilateral semidirect products of transformation semigroups