Abstract
In this note, we prove an existence and approximation result for a class of state-dependent rate-independent problems (which have already been investigated in [6]), by passing to the limit in a time-discretization scheme with suitably constructed variable-time steps.
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References
E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984), 570–598.
M. Brokate, P. Krejčí, and H. Schnabel, On uniqueness in evolution quasivariational inequalities. J. Convex Analysis 11 (2004), 111–130.
J. Diestel and J.J. Uhl, Jr., Vector Measures. American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 15.
A. Mielke, Evolution in rate-independent systems. In In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations. Evolutionary Equations, volume 2, pages 461–559, Elsevier B.V., 2005.
A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis. In H.-D. Alber, R. Balean, and R. Farwig, editors, Proceedings of the Workshop on “Models of Continuum Mechanics in Analysis and Engineering”, pages 117–129. Shaker-Verlag, 1999.
A. Mielke and R. Rossi, Existence and uniqueness results for general rate-independent. problems. Quaderno del Seminario Matematico di Brescia n. 21/2005, 2005. Math. Models Methods Appl. Sci. (to appear).
A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA Nonl. Diff. Eqns. Appl. 11 (2004), 151–189.
A. Mielke, F. Theil, and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002), 137–177.
R. Rockafellar, Convex Analysis. Princeton University Press, 1970.
R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12 (2006), 564–614.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Rossi, R. (2006). Existence and Approximation Results for General Rate-independent Problems via a Variable Time-step Discretization Scheme. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_36
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DOI: https://doi.org/10.1007/978-3-7643-7719-9_36
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