Uniqueness of a quasivariational sweeping process on functions of bounded variation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, p. 363-394

We prove existence and uniqueness of a quasivariational sweeping process on functions of bounded variation thereby generalizing previous results for absolutely continuous functions. It turns out that the size of the discontinuities plays a crucial role: In case they are small enough we prove existence and uniqueness. For large jumps we present a counterexample to the uniqueness of the solution. Finally we show that the condition on the jump size can be replaced by suitable conditions on the shape of the convex set.

Published online : 2018-06-21
Classification:  49J40,  47J20,  34G25,  34C55
@article{ASNSP_2012_5_11_2_363_0,
     author = {Roche, Thomas},
     title = {Uniqueness of a quasivariational sweeping process on functions of bounded variation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {2},
     year = {2012},
     pages = {363-394},
     mrnumber = {3011995},
     zbl = {1250.49012},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_363_0}
}
Roche, Thomas. Uniqueness of a quasivariational sweeping process on functions of bounded variation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 363-394. http://www.numdam.org/item/ASNSP_2012_5_11_2_363_0/

[1] G. Aumann, “Reelle Funktionen”, Springer-Verlag, New York, 1954. | MR 61652 | Zbl 0181.05801

[2] M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal. 11 (2004), 111–130. | MR 2159467 | Zbl 1061.49006

[3] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media, J. Engrg. Mater. Tech. 99 (1977), 2–15.

[4] V. S. Kozjankin, M. A. Krasnosel’skiĭ and A. V. Pokrovskiĭ, Vibrationally stable hysterons, Soviet Math. Dokl. 13 (1972), 1305–1309. | Zbl 0268.93027

[5] M. A. Krasnosel’skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabrejko, E. A. Lifshĭts and A. V. Pokrovskiĭ, Hysterant operator, Soviet Math. Dokl. 11 (1970), 29–33. | Zbl 0212.58002

[6] M. A. Krasnosel’skiĭ and A. V. Pokrovskiĭ, “Sistemy s Gisterezisom”, Nauka, Moscow, 1983. | Zbl 1092.47508

[7] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, In: “Nonlinear Differential Equations” (Chvalatice, 1998), volume 404 of Chapman & Hall/CRC Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 1999, 47–110, | MR 1695378 | Zbl 0949.47053

[8] P. Krejčí, The Kurzweil integral with exclusion of negligible sets, Math. Bohem. 128 (2003), 277–292. | MR 2012605 | Zbl 1051.26006

[9] P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal. 9 (2002), 159–183. | MR 1917394 | Zbl 1001.49014

[10] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math. 54 (2009), 117–145. | MR 2491851 | Zbl 1212.49007

[11] P. Krejčí and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Contin. Dyn. Syst. Ser. B 15 (2011), 637–650. | MR 2774131 | Zbl 1214.49022

[12] M. Kunze and M. D. P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal. 5 (1997), 57–72. | MR 1451848 | Zbl 0880.34017

[13] M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal. 12 (1998), 179–191. | MR 1677798 | Zbl 0923.34018

[14] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 418–449. | MR 111875 | Zbl 0090.30002

[15] A. Mielke and R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems, Math. Models Methods Appl. Sci. 17 (2007), 81–123. | MR 2290410 | Zbl 1121.34052

[16] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 151–189. | MR 2210284 | Zbl 1061.35182

[17] M. D. P. Monteiro Marques, Rafle par un convexe semi-continu inférieurement d’intérieur non vide en dimension finie, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 307–310. | MR 761253 | Zbl 0582.49012

[18] J.-J. Moreau, Problème d’évolution associé à un convexe mobile d’un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A791–A794. | MR 367750 | Zbl 0248.35021

[19] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equation 26 (1977), 347–374. | MR 508661 | Zbl 0356.34067

[20] R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping processes, Adv. Differential Equations 10 (2005), 527–552. | MR 2134049 | Zbl 1110.34039

[21] H. Schnabel, “Zur Wohlgestelltheit des Gurson-Modells”, Doctoral Thesis, Technische Universität München, 2006.

[22] U. Stefanelli, Nonlocal quasivariational evolution problems, J. Differential Equations 229 (2006), 204–228. | MR 2265625 | Zbl 1114.34044