BV solutions of rate independent variational inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 269-315.

We prove a theorem providing a geometric characterization of BV continuous vector rate independent operators. We apply this theorem to rate independent evolution variational inequalities and deduce new continuity properties of their solution operator: the vectorial play operator.

Published online:
Classification: 49J40, 47J35, 74C15, 34C55, 26A45
Recupero, Vincenzo 1

1 Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italia
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Recupero, Vincenzo. $\protect \mathbf{BV}$ solutions of rate independent variational inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 269-315. http://archive.numdam.org/item/ASNSP_2011_5_10_2_269_0/

[1] C. D. Aliprantis and K. C. Border, “Infinite Dimensional Analysis” (Third Edition), Springer, Berlin, Heidelberg, 2006. | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. | MR | Zbl

[3] N. Bourbaki, “Fonctions d’une variable rèelle”, Hermann, Paris, 1958. | MR

[4] H. Brezis, “Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert”, North-Holland Mathematical Studies, Vol. 5, North-Holland Publishing Company, Amsterdam, 1973. | MR | Zbl

[5] H. Brezis, “Analyse fonctionelle - Théorie et applications”, Masson, Paris, 1983. | MR | Zbl

[6] M. Brokate and J. Sprekels, “Hysteresis and Phase Transitions”, Applied Mathematical Sciences, Vol. 121, Springer-Verlag, New York, 1996. | MR | Zbl

[7] G. Dal Maso, P. Le Floch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483–548. | MR | Zbl

[8] N. Dinculeanu, “Vector Measures”, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Berlin, 1967. | MR | Zbl

[9] J. L. Doob, “Measure Theory”, Springer-Verlag, New York, 1994. | MR | Zbl

[10] N. Dunford and J. Schwartz, “Linear Operators, Part 1”, Wiley Interscience, New York, 1958. | MR | Zbl

[11] H. Federer, “Geometric Measure Theory”, Springer-Verlag, Berlin-Heidelberg, 1969. | MR | Zbl

[12] M. A. Krasnosel’skiǐ and A. V. Pokrovskiǐ, “Systems with Hysteresis”, Springer-Verlag, Berlin Heidelberg, 1989. | MR

[13] P. Krejčí, Vector hysteresis models, European J. Appl. Math. 2 (1991), 281–292. | MR | Zbl

[14] P. Krejčí, “Hysteresis, Convexity and Dissipation in Hyperbolic Equations”, Gakuto International Series Mathematical Sciences and Applications, Vol. 8, Gakkōtosho, Tokyo, 1996. | MR | Zbl

[15] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, In: “Nonlinear Differential Equations” (Chvalatice, 1998), Vol. 404, Chapman & Hall/CRC Res. Notes Math., 1999, 47–110. | MR | Zbl

[16] P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal. 9 (2002), 159–183. | MR | Zbl

[17] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math. 54 (2009), 117–145. | EuDML | MR | Zbl

[18] S. Lang, “Real and Functional Analysis” (Third Edition), Graduate Text in Mathematics, Vol. 142, Springer Verlag, New York, 1993. | MR | Zbl

[19] I. D. Mayergoyz, “Mathematical Models of Hysteresis”, Springer-Verlag, New York, 1991. | MR | Zbl

[20] A. Mielke, Evolution in rate-independent systems, In: “Handbook of Differential Equations, Evolutionary Equations”, Vol. 2, C. Dafermos and E. Fereisl (eds.), Elsevier, 2005, 461–559. | MR | Zbl

[21] V. Recupero, BV-extension of rate independent operators, Math. Nachr. 282 (2009), 86–98. | MR | Zbl

[22] V. Recupero, On locally isotone rate independent operators, Appl. Math. Letters. 20 (2007), 1156–1160. | MR | Zbl

[23] V. Recupero, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci. 31 (2008), 1283–1295. | MR | Zbl

[24] V. Recupero, Sobolev and strict continuity of general hysteresis operators, Math. Methods Appl. Sci. 32 (2009), 2003–2018. | MR | Zbl

[25] V. Recupero, On a class of scalar variational inequalities with measure data, Appl. Anal. 88 (2009), 1739–1753. | MR | Zbl

[26] U. Stefanelli, A variational characterization of rate independent evolution, Math. Nachr. 282 (2009), 1492–1512. | MR | Zbl

[27] A. Visintin, “Differential Models of Hysteresis”, Applied Mathematical Sciences, Vol. 111, Springer-Verlag, Berlin Heidelberg, 1994. | MR | Zbl