A continuity method for sweeping processes. (English) Zbl 1237.34116
The author considers the “sweeping process” in a Hilbert space \(H\), specifically, the problem
\[
-y'(t)\in N_{C(t)}(y(t)),\quad y(0)= y_0\in C(0),\quad y(t)\in C(t),\quad t\in [0,T],
\]
where \(C: [0,T]\to 2^H\) and \(N_{C(t)}(x)\) the exterior normal cone to \(C(t)\) at \(y(t)\). The goal of the paper is to extend results in the literature in which the moving convex set \(C\) is Lipschitz continuous [see J. J. Moreau, Rafle par un convexe variable. I. Trav. Semin. d’Anal. convexe, Montpellier, Vol. 1, Expose No. 15, 43 p. (1971; Zbl 0343.49019) and Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26, 347–374 (1977; Zbl 0356.34067)]. An existence and uniqueness result is proved in the space of mappings of bounded variation. Also the author proves an extension theorem for a rate independent operator. This result is used for proving the continuous dependence of the sweeping process on initial data.
Reviewer: Abdelghani Ouahab (Sidi Bel Abbes)
References:
| [1] | Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis (2006), Springer: Springer Berlin, Heidelberg · Zbl 1156.46001 |
| [2] | Ambrosio, L.; Gigli, N.; Savarè, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures (2005), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1090.35002 |
| [3] | Brezis, H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., vol. 5 (1973), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam · Zbl 0252.47055 |
| [4] | Brokate, M.; Sprekels, J., Hysteresis and Phase Transitions, Appl. Math. Sci., vol. 121 (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0951.74002 |
| [5] | Dinculeanu, N., Vector Measures, Int. Ser. Monogr. Pure Appl. Math., vol. 95 (1967), Pergamon Press: Pergamon Press Berlin |
| [6] | Doob, J. L., Measure Theory (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0791.28001 |
| [7] | Edmond, J. F.; Thibault, L., BV solutions of nonconvex sweeping process differential inclusions with perturbations, J. Differential Equations, 226, 135-179 (2006) · Zbl 1110.34038 |
| [8] | Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0176.00801 |
| [9] | Gariepy, R. F.; Ziemer, W. P., Modern Real Analysis (1995), PWS Publishing Company: PWS Publishing Company Boston |
| [10] | Kenmochi, N., Solvability of nonlinear evolutions equations with time-dependent constraints and applications, Bull. Fac. Educ., Chiba Univ., 30, 1-87 (1981) · Zbl 0662.35054 |
| [11] | Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 8 (1997), Gakkōtosho: Gakkōtosho Tokyo |
| [12] | Krejčí, P.; Laurençot, P., Generalized variational inequalities, J. Convex Anal., 9, 159-183 (2002) · Zbl 1001.49014 |
| [13] | Krejčí, P.; Liero, M., Rate independent Kurzweil processes, Appl. Math., 54, 117-145 (2009) · Zbl 1212.49007 |
| [14] | Lang, S., Real and Functional Analysis, Grad. Texts in Math., vol. 142 (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0831.46001 |
| [15] | Monteiro Marques, M. D.P., Differential Inclusions in Nonsmooth Mechanical Problems - Shocks and Dry Friction (1993), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0802.73003 |
| [16] | Moreau, J. J., Rafle par un convexe variable, Sém. Anal. Convexe Montpellier, 1 (1971), Exposé No. 14, 43 pp · Zbl 0343.49019 |
| [17] | Moreau, J. J., Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26, 347-374 (1977) · Zbl 0356.34067 |
| [18] | Recupero, V., The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31, 1283-1295 (2008) · Zbl 1140.74021 |
| [19] | Recupero, V., BV solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10, 269-315 (2011) · Zbl 1229.49012 |
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