Abstract

Many nonequilibrium rate independent processes arising in elastoplasticity, ferromagnetism and phase transitions are described by an evolution variational inequality with a convex constraint in a Hilbert space. The resulting solution operator is called "play operator" and acts on absolutely continuous functions. For nonregular data two natural notions of BV solutions have been proposed by the authors, giving rise to different extensions of the play operator to BV. We prove that these extensions are equal if and only if the convex constraint is a non-obtuse polyhedron.

Author information

Contact details are reproduced from the original publication and may be historical.

Pavel Krejčí

Institute of Mathematics, Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic

krejci@math.cas.cz

Vincenzo Recupero

Dip. di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy

vincenzo.recupero@polito.it

Keywords

  • Variational inequalities
  • rate independence
  • convex sets

Mathematics Subject Classification

47J20, 74C05

References

  1. H. Brézis: Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematical Studies 5, North-Holland, Amsterdam (1973).Google Scholar
  2. M. Brokate, J. Sprekels: Hysteresis and Phase Transitions, Applied Mathematical Sciences 121, Springer, New York (1996).https://doi.org/10.1007/978-1-4612-4048-8Google Scholar
  3. H. Federer: Geometric Measure Theory, Springer, Berlin (1969).Google Scholar
  4. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms I, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  5. M. A. Krasnosel’skiǐ, A. V. Pokrovskiǐ: Systems with Hysteresis, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-61302-9Google Scholar
  6. P. Krejčí: Vector hysteresis models, Eur. J. Appl. Math. 2 (1991) 281–292.https://doi.org/10.1017/S0956792500000541Google Scholar
  7. P. Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series Mathematical Sciences and Applications 8, Gakkōtosho, Tokyo (1996).Google Scholar
  8. P. Krejčí: Evolution variational inequalities and multidimensional hysteresis operators, in: Nonlinear Differential Equations (Chvalatice, 1998), P. Drábek et al. (ed.), Chapman & Hall/CRC Res. Notes Math. 404, Chapman & Hall/CRC, Boca Raton (1999) 47–110.https://doi.org/10.1201/9780429332555-2Google Scholar
  9. P. Krejčí, P. Laurençot: Generalized variational inequalities, J. Convex Analysis 9 (2002) 159–183.https://doi.org/10.68381/jca09008Google Scholar
  10. P. Krejčí, M. Liero: Rate independent Kurzweil processes, Appl. Math., Praha 54 (2009) 117–145.https://doi.org/10.1007/s10492-009-0009-5Google Scholar
  11. P. Krejčí, T. Roche: Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Contin. Dyn. Syst., Ser. B 15 (2011) 637–650.https://doi.org/10.3934/dcdsb.2011.15.637Google Scholar
  12. J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957) 418–449.https://doi.org/10.21136/CMJ.1957.100258Google Scholar
  13. A. Mielke: Evolution in rate-independent systems, in: Handbook of Differential Equations: Evolutionary Equations. Vol. II, C. Dafermos, E. Feireisl (eds.), Elsevier, Amsterdam (2005) 461–559.https://doi.org/10.1016/S1874-5717(06)80009-5Google Scholar
  14. J. J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  15. V. Recupero: On locally isotone rate independent operators, Appl. Math. Lett. 20 (2007) 1156–1160.https://doi.org/10.1016/j.aml.2006.10.006Google Scholar
  16. V. Recupero: BV-extension of rate independent operators, Math. Nachr. 282 (2009) 86–98.https://doi.org/10.1002/mana.200610723Google Scholar
  17. V. Recupero: On a class of scalar variational inequalities with measure data, Appl. Anal. 88 (2009) 1739–1753.https://doi.org/10.1080/00036810903397446Google Scholar
  18. V. Recupero: BV solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10 (2011) 269–315.https://doi.org/10.2422/2036-2145.2011.2.02Google Scholar
  19. V. Recupero: Extending vector hysteresis operators, J. Phys.: Conf. Ser. 268 (2011) 0120124.https://doi.org/10.1088/1742-6596/268/1/012024Google Scholar
  20. A. Visintin: Differential Models of Hysteresis, Applied Mathematical Sciences 111, Springer, Berlin (1994).https://doi.org/10.1007/978-3-662-11557-2Google Scholar

Suggested citation

P. Krejčí, V. Recupero. “Comparing BV Solutions of Rate Independent Processes.” Journal of Convex Analysis 21 (2014), No. 1, 121–146. https://doi.org/10.68381/jca21006

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.