Local Lipschitz continuity of the stop operator

Wolfgang Desch

Applications of Mathematics (1998)

  • Volume: 43, Issue: 6, page 461-477
  • ISSN: 0862-7940

Abstract

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On a closed convex set Z in ℝ N with sufficiently smooth ( 𝒲 2 , ∞ ) boundary, the stop operator is locally Lipschitz continuous from 𝐖 1 , 1 ( [ 0 , T ] , ℝ N ) × Z into 𝐖 1 , 1 ( [ 0 , T ] , ℝ N ) . The smoothness of the boundary is essential: A counterexample shows that 𝒞 1 -smoothness is not sufficient.

How to cite

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Desch, Wolfgang. "Local Lipschitz continuity of the stop operator." Applications of Mathematics 43.6 (1998): 461-477. <http://eudml.org/doc/33021>.

@article{Desch1998,
abstract = {On a closed convex set $Z$ in $\{\mathbb \{R\}\}^N$ with sufficiently smooth ($\{\mathcal \{W\}\}^\{2,\infty \}$) boundary, the stop operator is locally Lipschitz continuous from $\{\mathbf \{W\}\}^\{1,1\}([0,T],\{\mathbb \{R\}\}^N) \times Z$ into $\{\mathbf \{W\}\}^\{1,1\}([0,T],\{\mathbb \{R\}\}^N)$. The smoothness of the boundary is essential: A counterexample shows that $\{\mathcal \{C\}\}^1$-smoothness is not sufficient.},
author = {Desch, Wolfgang},
journal = {Applications of Mathematics},
keywords = {hysteresis; stop operator; differential inclusion; Lipschitz continuity; hysteresis; stop operator; differential inclusions; Lipschitz continuity},
language = {eng},
number = {6},
pages = {461-477},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Local Lipschitz continuity of the stop operator},
url = {http://eudml.org/doc/33021},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Desch, Wolfgang
TI - Local Lipschitz continuity of the stop operator
JO - Applications of Mathematics
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 43
IS - 6
SP - 461
EP - 477
AB - On a closed convex set $Z$ in ${\mathbb {R}}^N$ with sufficiently smooth (${\mathcal {W}}^{2,\infty }$) boundary, the stop operator is locally Lipschitz continuous from ${\mathbf {W}}^{1,1}([0,T],{\mathbb {R}}^N) \times Z$ into ${\mathbf {W}}^{1,1}([0,T],{\mathbb {R}}^N)$. The smoothness of the boundary is essential: A counterexample shows that ${\mathcal {C}}^1$-smoothness is not sufficient.
LA - eng
KW - hysteresis; stop operator; differential inclusion; Lipschitz continuity; hysteresis; stop operator; differential inclusions; Lipschitz continuity
UR - http://eudml.org/doc/33021
ER -

References

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  7. 10.1017/S0956792500000541, Euro. J. of Applied Math. 2 (1991), 281–292. (1991) MR1123144DOI10.1017/S0956792500000541
  8. Hysteresis, Convexity, and Dissipation in Hyperbolic Equations, Gakkotosho, Tokyo, 1996. (1996) MR2466538
  9. Evolution variational inequalities and multidimensional hysteresis operators, Manuscript. 
  10. Continuity of hysteresis operators in Sobolev spaces, Appl. Math. 35 (1990), 60–66. (1990) MR1039411
  11. Differential Models of Hysteresis, Springer, Berlin, 1994. (1994) Zbl0820.35004MR1329094

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