# Article

Full entry | PDF   (0.3 MB)
Keywords:
hysteresis; stop operator; differential inclusion; Lipschitz continuity
Summary:
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.
References:
[1] M. Berger and B. Gostiaux: Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics 115, Springer, New York, 1988. MR 0917479
[2] M. Brokate and P. Krejčí: Wellposedness of kinematic hardening models in elastoplasticity. Christian-Albrechts-Universität Kiel, Berichtsreihe des Mathematischen Seminars Kiel, Bericht 96–4, Februar 1996.
[3] M. Brokate and J. Sprekels: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121, Springer, New York, 1996. MR 1411908
[4] W. Desch and J. Turi: The stop operator related to a convex polyhedron. Manuscript.
[5] J. Dieudonné: Foundations of Modern Analysis. Academic Press, New York, London, 1969. MR 0349288
[6] M. A. Krasnosel’skii and A. V. Pokrovskii: Systems with Hysteresis. Springer, Berlin, 1989. MR 0987431
[7] P. Krejčí: Vector hysteresis models. Euro. J. of Applied Math. 2 (1991), 281–292. DOI 10.1017/S0956792500000541 | MR 1123144
[8] P. Krejčí: Hysteresis, Convexity, and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo, 1996. MR 2466538
[9] P. Krejčí: Evolution variational inequalities and multidimensional hysteresis operators. Manuscript.
[10] P. Krejčíand V. Lovicar: Continuity of hysteresis operators in Sobolev spaces. Appl. Math. 35 (1990), 60–66. MR 1039411
[11] A. Visintin: Differential Models of Hysteresis. Springer, Berlin, 1994. MR 1329094 | Zbl 0820.35004

Partner of
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.86551433801651, "perplexity": 2936.381427861314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948542031.37/warc/CC-MAIN-20171214074533-20171214094533-00053.warc.gz"}