Uniqueness of a quasivariational sweeping process on functions of bounded variation

Roche, Thomas

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 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.

Classification : 49J40, 47J20, 34G25, 34C55

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

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

[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

[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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl