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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

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 & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637
References:
[1]

G. Aumann, "Reelle Funktionen,", Springer-Verlag, (1954).   Google Scholar

[2]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. Convex Anal., 11 (2004), 111.   Google Scholar

[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.  doi: 10.1115/1.3443401.  Google Scholar

[4]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[5]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[6]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter,, Czechoslovak Math. J., 7(82) (1957), 418.   Google Scholar

[7]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.   Google Scholar

[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).   Google Scholar

[9]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Eq., 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[10]

V. Recupero, On locally isotone rate independent operators,, Applied Mathematics Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[11]

V. Recupero, $BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[12]

V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492.  doi: 10.1002/mana.200810803.  Google Scholar

[14]

M. Tvrdý, Regulated functions and the Perron-Stieltjes integral,, Čas. pěst. Mat., 114 (1989), 187.   Google Scholar

show all references

References:
[1]

G. Aumann, "Reelle Funktionen,", Springer-Verlag, (1954).   Google Scholar

[2]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. Convex Anal., 11 (2004), 111.   Google Scholar

[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.  doi: 10.1115/1.3443401.  Google Scholar

[4]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[5]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[6]

J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter,, Czechoslovak Math. J., 7(82) (1957), 418.   Google Scholar

[7]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.   Google Scholar

[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).   Google Scholar

[9]

J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Eq., 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[10]

V. Recupero, On locally isotone rate independent operators,, Applied Mathematics Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[11]

V. Recupero, $BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[12]

V. Recupero, $BV$-solutions of rate independent variational inequalities,, To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492.  doi: 10.1002/mana.200810803.  Google Scholar

[14]

M. Tvrdý, Regulated functions and the Perron-Stieltjes integral,, Čas. pěst. Mat., 114 (1989), 187.   Google Scholar

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