Continuity of the non-convex play operator in the space of rectifiable curves

Jana Kopfová; Vincenzo Recupero

Applications of Mathematics (2023)

  • Volume: 68, Issue: 6, page 727-750
  • ISSN: 0862-7940

Abstract

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We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the B V -norm and to the B V -strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent.

How to cite

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Kopfová, Jana, and Recupero, Vincenzo. "Continuity of the non-convex play operator in the space of rectifiable curves." Applications of Mathematics 68.6 (2023): 727-750. <http://eudml.org/doc/299404>.

@article{Kopfová2023,
abstract = {We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the $\{BV\}$-norm and to the $\{BV\}$-strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent.},
author = {Kopfová, Jana, Recupero, Vincenzo},
journal = {Applications of Mathematics},
keywords = {evolution variational inequalities; play operator; sweeping processes; functions of bounded variation; prox-regular set},
language = {eng},
number = {6},
pages = {727-750},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuity of the non-convex play operator in the space of rectifiable curves},
url = {http://eudml.org/doc/299404},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Kopfová, Jana
AU - Recupero, Vincenzo
TI - Continuity of the non-convex play operator in the space of rectifiable curves
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 6
SP - 727
EP - 750
AB - We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the ${BV}$-norm and to the ${BV}$-strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent.
LA - eng
KW - evolution variational inequalities; play operator; sweeping processes; functions of bounded variation; prox-regular set
UR - http://eudml.org/doc/299404
ER -

References

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  1. Benabdellah, H., 10.1006/jdeq.1999.3756, J. Differ. Equations 164 (2000), 286-295. (2000) Zbl0957.34061MR1765576DOI10.1006/jdeq.1999.3756
  2. Bounkhel, M., Thibault, L., Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2005), 359-374. (2005) Zbl1086.49016MR2159846
  3. Brézis, H., 10.1016/s0304-0208(08)x7125-7, North-Holland Mathematical Studies 5. North-Holland, Amsterdam (1973), French. (1973) Zbl0252.47055MR0348562DOI10.1016/s0304-0208(08)x7125-7
  4. Brokate, M., Krejčí, P., Schnabel, H., On uniqueness in evolution quasivariational inequalities, J. Convex Anal. 11 (2004), 111-130. (2004) Zbl1061.49006MR2159467
  5. Brokate, M., Sprekels, J., 10.1007/978-1-4612-4048-8, Applied Mathematical Sciences 121. Springer, New York (1996). (1996) Zbl0951.74002MR1411908DOI10.1007/978-1-4612-4048-8
  6. Castaing, C., Marques, M. D. P. Monteiro, Evolution problems associated with nonconvex closed moving sets with bounded variation, Port. Math. 53 (1996), 73-87. (1996) Zbl0848.35052MR1384501
  7. Clarke, F. H., Stern, R. J., Wolenski, P. R., Proximal smoothness and the lower- C 2 property, J. Convex Anal. 2 (1995), 117-144. (1995) Zbl0881.49008MR1363364
  8. Colombo, G., Goncharov, V. V., 10.1023/A:1008774529556, Set-Valued Anal. 7 (1999), 357-374. (1999) Zbl0957.34060MR1756914DOI10.1023/A:1008774529556
  9. Colombo, G., Marques, M. D. P. Monteiro, 10.1016/S0022-0396(02)00021-9, J. Differ. Equations 187 (2003), 46-62. (2003) Zbl1029.34052MR1946545DOI10.1016/S0022-0396(02)00021-9
  10. Colombo, G., Thibault, L., Prox-regular sets and applications, Handbook of Nonconvex Analysis and Applications International Press, Somerville (2010), 99-182. (2010) Zbl1221.49001MR2768810
  11. J. Diestel, J. J. Uhl, Jr., 10.1090/surv/015, Mathematical Surveys 15. AMS, Providence (1977). (1977) Zbl0369.46039MR0453964DOI10.1090/surv/015
  12. Dinculeanu, N., Vector Measures, International Series of Monographs in Pure and Applied Mathematics 95. Pergamon Press, Berlin (1967). (1967) Zbl0142.10502MR0206190
  13. Edmond, J. F., Thibault, L., 10.1016/j.jde.2005.12.005, J. Differ. Equations 226 (2006), 135-179. (2006) Zbl1110.34038MR2232433DOI10.1016/j.jde.2005.12.005
  14. Federer, H., 10.1090/S0002-9947-1959-0110078-1, Trans. Am. Math. Soc. 93 (1959), 418-491. (1959) Zbl0089.38402MR0110078DOI10.1090/S0002-9947-1959-0110078-1
  15. Federer, H., 10.1007/978-3-642-62010-2, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153. Springer, Berlin (1969). (1969) Zbl0176.00801MR0257325DOI10.1007/978-3-642-62010-2
  16. Klein, O., Recupero, V., 10.1088/1742-6596/727/1/012006, J. Phys., Conf. Ser. 727 (2016), Article ID 012006, 12 pages. (2016) Zbl1366.46062MR3566835DOI10.1088/1742-6596/727/1/012006
  17. Kopfová, J., Recupero, V., 10.1016/j.jde.2016.08.025, J. Differ. Equations 261 (2016), 5875-5899. (2016) Zbl1354.34112MR3548274DOI10.1016/j.jde.2016.08.025
  18. Krasnosel'skij, M. A., Pokrovskij, A. V., 10.1007/978-3-642-61302-9, Springer, Berlin (1989). (1989) Zbl0665.47038MR0987431DOI10.1007/978-3-642-61302-9
  19. Krejčí, P., 10.1017/S0956792500000541, Eur. J. Appl. Math. 2 (1991), 281-292. (1991) Zbl0754.73015MR1123144DOI10.1017/S0956792500000541
  20. Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO International Series. Mathematical Sciences and Applications 8. Gakkotosho, Tokyo (1996). (1996) Zbl1187.35003MR2466538
  21. Krejčí, P., Monteiro, G. A., Recupero, V., 10.1007/s00245-021-09801-8, Appl. Math. Optim. 84, Suppl. 2 (2021), 1477-1504. (2021) Zbl1495.34087MR4356901DOI10.1007/s00245-021-09801-8
  22. Krejčí, P., Monteiro, G. A., Recupero, V., 10.3934/cpaa.2022087, Commun. Pure Appl. Anal. 21 (2022), 2999-3029. (2022) Zbl07606668MR4484114DOI10.3934/cpaa.2022087
  23. Krejčí, P., Roche, T., 10.3934/dcdsb.2011.15.637, Discrete Contin. Dyn. Syst., Ser. B 15 (2011), 637-650. (2011) Zbl1214.49022MR2774131DOI10.3934/dcdsb.2011.15.637
  24. Lang, S., 10.1007/978-1-4612-0897-6, Graduate Texts in Mathematics 142. Springer, New York (1993). (1993) Zbl0831.46001MR1216137DOI10.1007/978-1-4612-0897-6
  25. Mayergoyz, I. D., 10.1007/978-1-4612-3028-1, Springer, New York (1991). (1991) Zbl0723.73003MR1083150DOI10.1007/978-1-4612-3028-1
  26. Mielke, A., Roubíček, T., 10.1007/978-1-4939-2706-7, Applied Mathematical Sciences 193. Springer, New York (2015). (2015) Zbl1339.35006MR3380972DOI10.1007/978-1-4939-2706-7
  27. Poliquin, R. A., Rockafellar, R. T., Thibault, L., 10.1090/S0002-9947-00-02550-2, Trans. Am. Math. Soc. 352 (2000), 5231-5249. (2000) Zbl0960.49018MR1694378DOI10.1090/S0002-9947-00-02550-2
  28. Recupero, V., 10.1002/mma.968, Math. Methods Appl. Sci. 31 (2008), 1283-1295. (2008) Zbl1140.74021MR2431427DOI10.1002/mma.968
  29. Recupero, V., 10.1002/mma.1124, Math. Methods Appl. Sci. 32 (2009), 2003-2018. (2009) Zbl1214.47081MR2560936DOI10.1002/mma.1124
  30. Recupero, V., 10.2422/2036-2145.2011.2.02, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10 (2011), 269-315. (2011) Zbl1229.49012MR2856149DOI10.2422/2036-2145.2011.2.02
  31. Recupero, V., 10.1016/j.jde.2015.05.019, J. Differ. Equations 259 (2015), 4253-4272. (2015) Zbl1322.49014MR3369277DOI10.1016/j.jde.2015.05.019
  32. Recupero, V., Sweeping processes and rate independence, J. Convex Anal. 23 (2016), 921-946. (2016) Zbl1357.34103MR3533181
  33. Recupero, V., Convex valued geodesics and applications to sweeping processes with bounded retraction, J. Convex Anal. 27 (2020), 535-556. (2020) Zbl1444.34075MR4058035
  34. Venel, J., 10.1007/s00211-010-0329-0, Numer. Math. 118 (2011), 367-400. (2011) Zbl1222.65064MR2800713DOI10.1007/s00211-010-0329-0
  35. Vial, J.-P., 10.1287/moor.8.2.231, Math. Oper. Res. 8 (1983), 231-259. (1983) Zbl0526.90077MR0707055DOI10.1287/moor.8.2.231
  36. Visintin, A., 10.1007/978-3-662-11557-2, Applied Mathematical Sciences 111. Springer, Berlin (1994). (1994) Zbl0820.35004MR1329094DOI10.1007/978-3-662-11557-2

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