A remark on the local Lipschitz continuity of vector hysteresis operators

Pavel Krejčí

Applications of Mathematics (2001)

  • Volume: 46, Issue: 1, page 1-11
  • ISSN: 0862-7940

Abstract

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It is known that the vector stop operator with a convex closed characteristic Z of class C 1 is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary ∂ Z of Z . We prove that in the regular case, this condition is also necessary.

How to cite

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Krejčí, Pavel. "A remark on the local Lipschitz continuity of vector hysteresis operators." Applications of Mathematics 46.1 (2001): 1-11. <http://eudml.org/doc/33074>.

@article{Krejčí2001,
abstract = {It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^1$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.},
author = {Krejčí, Pavel},
journal = {Applications of Mathematics},
keywords = {variational inequality; hysteresis operators; variational inequality; hysteresis operators},
language = {eng},
number = {1},
pages = {1-11},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A remark on the local Lipschitz continuity of vector hysteresis operators},
url = {http://eudml.org/doc/33074},
volume = {46},
year = {2001},
}

TY - JOUR
AU - Krejčí, Pavel
TI - A remark on the local Lipschitz continuity of vector hysteresis operators
JO - Applications of Mathematics
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 46
IS - 1
SP - 1
EP - 11
AB - It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^1$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.
LA - eng
KW - variational inequality; hysteresis operators; variational inequality; hysteresis operators
UR - http://eudml.org/doc/33074
ER -

References

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  1. Materials with Memory. Lecture Notes in Mathematics, Vol. 1682, Springer-Verlag, Berlin-Heidelberg, 1998. (1998) MR1619546
  2. 10.1016/S0921-4526(97)00319-0, Physica B 233 (1997), 342–347. (1997) DOI10.1016/S0921-4526(97)00319-0
  3. Opérateurs Maximaux Monotones, North-Holland Math. Studies, Amsterdam, 1973. (1973) 
  4. Elastoplastic constitutive laws of nonlinear kinematic hardening type, In: Functional Analysis with Current Applications in Science, Technology and Industry (M. Brokate, A. H. Siddiqi, eds.), Pitman Res. Notes Math. Ser., 377, Longman, Harlow, 1998, pp. 238–272. (1998) Zbl0911.73021MR1607891
  5. 10.1051/m2an/1998320201771, Math. Modelling Numer. Anal. 32 (1998), 177–209. (1998) MR1622607DOI10.1051/m2an/1998320201771
  6. 10.1023/A:1023221405455, Appl. Math. 43 (1998), 461–477. (1998) Zbl0937.47058MR1652108DOI10.1023/A:1023221405455
  7. 10.1006/jdeq.1998.3601, J. Differential Equations 157 (1999), 329–347. (1999) MR1710220DOI10.1006/jdeq.1998.3601
  8. 10.1080/17442509108833688, Stochastics Stochastics Rep. 35 (1991), 31–62. (1991) MR1110990DOI10.1080/17442509108833688
  9. Dynamical systems and variational inequalities, Ann. Oper. Res. 44 (1993), 9–42. (1993) MR1246835
  10. Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976, French edition: Dunod, Paris, 1972. (1976) MR0521262
  11. Systems with Hysteresis, Springer-Verlag, Berlin, 1989, Russian edition: Nauka, Moscow, 1983. (1989) MR0742931
  12. Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakuto Int. Ser. Math. Sci. Appl., Vol. 8, Gakkōtosho, Tokyo, 1996. (1996) MR2466538
  13. Evolution variational inequalities and multidimensional hysteresis operators, In: Nonlinear Differential Equations (P. Drábek, P. Krejčí and P. Takáč, eds.). Research Notes in Mathematics, Vol. 404, Chapman & Hall/CRC, London, 1999, pp. 47–110. (1999) MR1695378
  14. Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Eqations 26 (1977), 347–374. (1977) MR0508661
  15. Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction, Elsevier, Amsterdam, 1981. (1981) MR0600655
  16. Differential Models of Hysteresis, Springer-Verlag, Berlin-Heidelberg, 1994. (1994) Zbl0820.35004MR1329094

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