Abstract
The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in BV spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
G. Aumann: Reelle Funktionen. Springer-Verlag, Berlin-Götingen-Heidelberg, 1954.
M. Brokate, P. Krejčí, H. Schnabel: On uniqueness in evolution quasivariational inequalities. J. Convex Anal. 11 (2004), 111–130.
P. Drábek, P. Krejčí, P. Takáč: Nonlinear Differential Equations. Research Notes in Mathematics, Vol. 404. Chapman & Hall/CRC, London, 1999.
M. A. Krasnosel’skii, A. V. Pokrovskii: Systems with Hysteresis. Nauka, Moscow, 1983. (In Russian; English edition Springer 1989.)
P. Krejčí, J. Kurzweil: A nonexistence result for the Kurzweil integral. Math. Bohem. 127 (2002), 571–580.
P. Krejčí, Ph. Laurençot: Generalized variational inequalities. J. Convex Anal. 9 (2002), 159–183.
P. Krejčí: The Kurzweil integral with exclusion of negligible sets. Math. Bohem. 128 (2003), 277–292.
J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7(82) (1957), 418–449.
A. Mielke, R. Rossi: Existence and uniqueness results for a class of rate-independent hysteresis problems. Math. Models Methods Appl. Sci. 17 (2007), 81–123.
A. Mielke, F. Theil: On rate-independent hysteresis models. NoDEA, Nonlinear Differ. Equ. Appl. 11 (2004), 151–189.
J.-J. Moreau: Evolution problem associated with a moving convex set in Hilbert space. J. Differ. Equations 26 (1977), 347–374.
R. T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, 1970.
Š. Schwabik: On a modified sum integral of Stieltjes type. Čas. Pěst. Mat. 98 (1973), 274–277.
Š. Schwabik: Generalized Ordinary Differential Equations. Series in Real Analysis, Vol. 5. World Scientific Publishing Co., Inc., River Edge, 1992.
M. Tvrdý: Regulated functions and the Perron-Stieltjes integral. Čas. Pěst. Mat. 114 (1989), 87–209.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krejčí, P., Liero, M. Rate independent Kurzweil processes. Appl Math 54, 117–145 (2009). https://doi.org/10.1007/s10492-009-0009-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-009-0009-5