Lipschitz continuous data dependence of sweeping processes in BV spaces

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December 2011, 15(3): 637-650. doi: 10.3934/dcdsb.2011.15.637

Lipschitz continuous data dependence of sweeping processes in BV spaces

Pavel Krejčí ^1, and Thomas Roche ^2,

1.	Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1
2.	Department of Mathematics / M6, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany

Received February 2010 Revised May 2010 Published February 2011

Abstract
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For a rate independent sweeping process with a time dependent smooth convex constraint, we prove that the Kurzweil solution for possibly discontinuous inputs depends locally Lipschitz continuously on the data in terms of the $BV$-norm.

Keywords: Rate independence, discontinuous sweeping process, Kurzweil integral..

Mathematics Subject Classification: Primary: 49J40; Secondary: 34C5.

Citation: Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637

References:

[1]	G. Aumann, "Reelle Funktionen," Springer-Verlag, New York, 1954.
[2]	M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130.
[3]	A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media, J. Engrg. Mater. Tech., 99 (1977), 2-15. doi: 10.1115/1.3443401.
[4]	P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[5]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[6]	J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7(82) (1957), 418-449.
[7]	A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
[8]	J.-J. Moreau, Problème d'évolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794.
[9]	J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Eq., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[10]	V. Recupero, On locally isotone rate independent operators, Applied Mathematics Letters, 20 (2007), 1156-1160. doi: 10.1016/j.aml.2006.10.006.
[11]	V. Recupero, $BV$-extension of rate independent operators, Math. Nachr., 282 (2009), 86-98. doi: 10.1002/mana.200610723.
[12]	V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().
[13]	U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803.
[14]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Čas. pěst. Mat., 114 (1989), 187-209.

show all references

References:

[1]	G. Aumann, "Reelle Funktionen," Springer-Verlag, New York, 1954.
[2]	M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130.
[3]	A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media, J. Engrg. Mater. Tech., 99 (1977), 2-15. doi: 10.1115/1.3443401.
[4]	P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[5]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[6]	J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7(82) (1957), 418-449.
[7]	A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
[8]	J.-J. Moreau, Problème d'évolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794.
[9]	J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Eq., 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[10]	V. Recupero, On locally isotone rate independent operators, Applied Mathematics Letters, 20 (2007), 1156-1160. doi: 10.1016/j.aml.2006.10.006.
[11]	V. Recupero, $BV$-extension of rate independent operators, Math. Nachr., 282 (2009), 86-98. doi: 10.1002/mana.200610723.
[12]	V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().
[13]	U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803.
[14]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Čas. pěst. Mat., 114 (1989), 187-209.

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