Abstract
On a closed convex set Z in ℝN with sufficiently smooth (W 2,∞) boundary, the stop operator is locally Lipschitz continuous from W 1,1([0,T]ℝN) × Z into W 1,1([0,T],ℝN). The smoothness of the boundary is essential: A counterexample shows that C 1-smoothness is not sufficient.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Berger and B. Gostiaux: Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics 115, Springer, New York, 1988.
M. Brokate and P. Krejčí: Wellposedness of kinematic hardening models in elastoplasticity. Christian-Albrechts-Universität Kiel, Berichtsreihe des Mathematischen Seminars Kiel, Bericht 96-4, February 1996.
M. Brokate and J. Sprekels: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121, Springer, New York, 1996.
W. Desch and J. Turi: The stop operator related to a convex polyhedron. Manuscript.
J. Dieudonné: Foundations of Modern Analysis. Academic Press, New York, London, 1969.
M. A. Krasnosel'skii and A. V. Pokrovskii: Systems with Hysteresis. Springer, Berlin, 1989.
P. Krejčí: Vector hysteresis models. Euro. J. of Applied Math. 2 (1991), 281–292.
P. Krejčí: Hysteresis, Convexity, and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo, 1996.
P. Krejčí: Evolution variational inequalities and multidimensional hysteresis operators. Manuscript.
P. Krejčí and V. Lovicar: Continuity of hysteresis operators in Sobolev spaces. Appl. Math. 35 (1990), 60–66.
A. Visintin: Differential Models of Hysteresis. Springer, Berlin, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Desch, W. Local Lipschitz continuity of the stop operator. Applications of Mathematics 43, 461–477 (1998). https://doi.org/10.1023/A:1023221405455
Issue Date:
DOI: https://doi.org/10.1023/A:1023221405455