zbMATH — the first resource for mathematics

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.

MSC:
 34G25 Evolution inclusions 47J30 Variational methods involving nonlinear operators
Full Text:
References:
 [1] Aliprantis, C.D.; Border, K.C., Infinite dimensional analysis, (2006), 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 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 Amsterdam · Zbl 0252.47055 [4] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, Appl. math. sci., vol. 121, (1996), Springer-Verlag New York · Zbl 0951.74002 [5] Dinculeanu, N., Vector measures, Int. ser. monogr. pure appl. math., vol. 95, (1967), Pergamon Press Berlin [6] Doob, J.L., Measure theory, (1994), 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 Berlin, Heidelberg · Zbl 0176.00801 [9] Gariepy, R.F.; Ziemer, W.P., Modern real analysis, (1995), 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) [11] Krejčí, P., Hysteresis, convexity and dissipation in hyperbolic equations, GAKUTO internat. ser. math. sci. appl., vol. 8, (1997), 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 New York · Zbl 0831.46001 [15] Monteiro Marques, M.D.P., Differential inclusions in nonsmooth mechanical problems - shocks and dry friction, (1993), 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.