BV solutions of rate independent differential inclusions

Pavel Krejčí; Vincenzo Recupero

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 607-619
  • ISSN: 0862-7959

Abstract

top
We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For BV (bounded variation) data we compare different notions of BV solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case we also give a geometric characterization of the cases when these kinds of solutions coincide for left continuous inputs.

How to cite

top

Krejčí, Pavel, and Recupero, Vincenzo. "$\rm BV$ solutions of rate independent differential inclusions." Mathematica Bohemica 139.4 (2014): 607-619. <http://eudml.org/doc/269846>.

@article{Krejčí2014,
abstract = {We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For $\rm BV$ (bounded variation) data we compare different notions of $\rm BV$ solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case we also give a geometric characterization of the cases when these kinds of solutions coincide for left continuous inputs.},
author = {Krejčí, Pavel, Recupero, Vincenzo},
journal = {Mathematica Bohemica},
keywords = {differential inclusion; stop operator; rate independence; convex set; differential inclusion; stop operator; rate independence; convex set},
language = {eng},
number = {4},
pages = {607-619},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\rm BV$ solutions of rate independent differential inclusions},
url = {http://eudml.org/doc/269846},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Krejčí, Pavel
AU - Recupero, Vincenzo
TI - $\rm BV$ solutions of rate independent differential inclusions
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 607
EP - 619
AB - We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For $\rm BV$ (bounded variation) data we compare different notions of $\rm BV$ solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case we also give a geometric characterization of the cases when these kinds of solutions coincide for left continuous inputs.
LA - eng
KW - differential inclusion; stop operator; rate independence; convex set; differential inclusion; stop operator; rate independence; convex set
UR - http://eudml.org/doc/269846
ER -

References

top
  1. Aumann, G., Reelle Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete 68 Springer, Berlin (1954), German. (1954) Zbl0056.05202MR0061652
  2. Bourbaki, N., Elements of Mathematics. Functions of a Real Variable. Elementary Theory, Springer Berlin (2004), translated from the 1976 French original. (2004) Zbl1085.26001MR2013000
  3. Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies 5 North-Holland, Amsterdam, Elsevier, New York French (1973). (1973) MR0348562
  4. Brokate, M., Sprekels, J., 10.1007/978-1-4612-4048-8_5, Applied Mathematical Sciences 121 Springer, New York (1996). (1996) Zbl0951.74002MR1411908DOI10.1007/978-1-4612-4048-8_5
  5. Fraňková, D., Regulated functions, Math. Bohem. 116 (1991), 20-59. (1991) MR1100424
  6. Klein, O., Representation of hysteresis operators acting on vector-valued monotaffine functions, Adv. Math. Sci. Appl. 22 (2012), 471-500. (2012) MR3100006
  7. Krasnosel'skiĭ, M. A., Pokrovskiĭ, A. V., Systems with Hysteresis, Springer Berlin (1989), translated from the Russian, Nauka, Moskva, 1983. (1989) MR0742931
  8. Krejčí, P., Evolution variational inequalities and multidimensional hysteresis operators, Nonlinear Differential Equations. Proceedings of the seminar in Differential Equations, Chvalatice, Czech Republic, 1998 Chapman & Hall/CRC Res. Notes Math. 404 Boca Raton (1999), 47-110 P. Drábek et al. (1999) Zbl0949.47053MR1695378
  9. Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series. Mathematical Sciences and Applications 8 Gakkōtosho, Tokyo (1996). (1996) MR2466538
  10. Krejčí, P., Laurençot, P., Generalized variational inequalities, J. Convex Anal. 9 (2002), 159-183. (2002) Zbl1001.49014MR1917394
  11. Krejčí, P., Laurençot, P., Hysteresis filtering in the space of bounded measurable functions, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 5 (2002), 755-772. (2002) Zbl1177.35125MR1934379
  12. Krejčí, P., Liero, M., 10.1007/s10492-009-0009-5, Appl. Math., Praha 54 (2009), 117-145. (2009) Zbl1212.49007MR2491851DOI10.1007/s10492-009-0009-5
  13. Krejčí, P., Recupero, V., Comparing BV solutions of rate independent processes, J. Convex Anal. 21 (2014), 121-146. (2014) MR3235307
  14. Krejčí, P., Roche, T., 10.3934/dcdsb.2011.15.637, Discrete Contin. Dyn. Syst., Ser. B 15 (2011), 637-650. (2011) MR2774131DOI10.3934/dcdsb.2011.15.637
  15. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957), 418-449. (1957) Zbl0090.30002MR0111875
  16. Logemann, H., Mawby, A. D., 10.1016/S0022-247X(03)00097-0, J. Math. Anal. Appl. 282 (2003), 107-127. (2003) Zbl1042.47049MR2000333DOI10.1016/S0022-247X(03)00097-0
  17. Marques, M. D. P. Monteiro, Differential Inclusions in Nonsmooth Mechanical Problems---Shocks and Dry Friction, Progress in Nonlinear Differential Equations and their Applications 9 Birkhäuser, Basel (1993). (1993) MR1231975
  18. Moreau, J. J., 10.1016/0022-0396(77)90085-7, J. Differ. Equations 26 (1977), 347-374. (1977) MR0508661DOI10.1016/0022-0396(77)90085-7
  19. Moreau, J. J., Solutions du processus de rafle au sens des mesures différentielles, Travaux Sém. Anal. Convexe 6 French (1976), Exposé No. 1, 17 pages. (1976) Zbl0369.49007MR0637888
  20. Moreau, J. J., Rafle par un convexe variable II, Travaux du Séminaire d'Analyse Convexe II U. É. R. de Math., Univ. Sci. Tech. Languedoc Montpellier French (1972), Exposé No. 3, 36 pages. Zbl0343.49020MR0637728
  21. Recupero, V., 10.1016/j.jde.2011.06.018, J. Differ. Equations 251 (2011), 2125-2142. (2011) Zbl1237.34116MR2823662DOI10.1016/j.jde.2011.06.018
  22. Recupero, V., BV solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10 (2011), 269-315. (2011) MR2856149
  23. Recupero, V., 10.1088/1742-6596/268/1/012024, J. Phys., Conf. Ser. 268 Article No. 0120124, 12 pages (2011). (2011) DOI10.1088/1742-6596/268/1/012024
  24. Recupero, V., 10.1002/mana.200610723, Math. Nachr. 282 (2009), 86-98. (2009) MR2473132DOI10.1002/mana.200610723
  25. Recupero, V., 10.1080/00036810903397446, Appl. Anal. 88 (2009), 1739-1753. (2009) Zbl1177.74308MR2588416DOI10.1080/00036810903397446
  26. Recupero, V., 10.1002/mma.1124, Math. Methods Appl. Sci. 32 (2009), 2003-2018. (2009) Zbl1214.47081MR2560936DOI10.1002/mma.1124
  27. Recupero, V., 10.1016/j.aml.2006.10.006, Appl. Math. Lett. 20 (2007), 1156-1160. (2007) Zbl1152.47059MR2358793DOI10.1016/j.aml.2006.10.006
  28. Siddiqi, A. H., Manchanda, P., Brokate, M., 10.1007/978-1-4613-0263-6_15, Trends in Industrial and Applied Mathematics 33, Amritsar, 2001 Appl. Optim. 72 Kluwer, Dordrecht (2002), 339-354 A. H. Siddiqi et al. (2002) MR1954114DOI10.1007/978-1-4613-0263-6_15
  29. Visintin, A., Differential Models of Hysteresis, Applied Mathematical Sciences 111 Springer, Berlin (1994), F. John et al. (1994) Zbl0820.35004MR1329094

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.