Local Lipschitz continuity of the stop operator

Wolfgang Desch

Applications of Mathematics (1998)

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

Abstract

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

How to cite

top

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

top
  1. Differential Geometry: Manifolds, Curves, and Surfaces, Graduate Texts in Mathematics 115, Springer, New York, 1988. (1988) MR0917479
  2. Wellposedness of kinematic hardening models in elastoplasticity, Christian-Albrechts-Universität Kiel, Berichtsreihe des Mathematischen Seminars Kiel, Bericht 96–4, Februar 1996. 
  3. Hysteresis and Phase Transitions, Applied Mathematical Sciences 121, Springer, New York, 1996. (1996) MR1411908
  4. The stop operator related to a convex polyhedron, Manuscript. 
  5. Foundations of Modern Analysis, Academic Press, New York, London, 1969. (1969) MR0349288
  6. Systems with Hysteresis, Springer, Berlin, 1989. (1989) MR0987431
  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

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.