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Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. (English) Zbl 1458.35439

Summary: We study a rate-independent system with non-convex energy in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized BV-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact.

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
49J40 Variational inequalities
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
35Q74 PDEs in connection with mechanics of deformable solids
35D40 Viscosity solutions to PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
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[1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. · Zbl 1090.35002
[2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Mathematics and its Applications, 10. D. Reidel Publishing Co., Dordrecht, Editura Academiei Republicii Socialiste Romania, Bucharest, 1986. · Zbl 0594.49001
[3] J. Dieudonné, Foundations of Modern Analysis. Enlarged and Corrected Printing, Pure and Applied Mathematics, Vol. 10-Ⅰ. Academic Press, New York-London, 1969. · Zbl 0176.00502
[4] M. A. Efendiev; A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13, 151-167 (2006) · Zbl 1109.74040
[5] P. Krejčí; P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9, 159-183 (2002) · Zbl 1001.49014
[6] P. Krejčí; M. Liero, Rate independent Kurzweil processes, Appl. Math., 54, 117-145 (2009) · Zbl 1212.49007
[7] D. Knees, Convergence analysis of time-discretization schemes for rate-independent systems, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 65, 38 pp. · Zbl 1437.49009
[8] P. Krejčí; V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21, 121-146 (2014) · Zbl 1305.47042
[9] D. Knees and S. Thomas, Optimal Control of a Rate-Independent System Constrained to Parametrized Balanced Viscosity Solutions, University of Kassel, 2018, arXiv: 1810.12572.
[10] G. Leoni, A First Course in Sobolev Spaces, Second edition, Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017. · Zbl 1382.46001
[11] A. Mainik; A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differ. Equ., 22, 73-99 (2005) · Zbl 1161.74387
[12] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26, 347-374 (1977) · Zbl 0356.34067
[13] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. · Zbl 1339.35006
[14] A. Mielke; R. Rossi; G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25, 585-615 (2009) · Zbl 1170.49036
[15] A. Mielke; R. Rossi; G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18, 36-80 (2012) · Zbl 1250.49041
[16] A. Mielke; R. Rossi; G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80, 381-410 (2012) · Zbl 1255.49078
[17] A. Mielke; R. Rossi; G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. (JEMS), 18, 2107-2165 (2016) · Zbl 1357.35007
[18] A. Mielke; S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13, 67-135 (2014) · Zbl 1295.35036
[19] V. Recupero, \(BV\) solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10, 269-315 (2011) · Zbl 1229.49012
[20] V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23, 921-946 (2016) · Zbl 1357.34103
[21] R. Rossi; A. Mielke; G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 7, 97-169 (2008) · Zbl 1183.35164
[22] V. Recupero; F. Santambrogio, Sweeping processes with prescribed behavior on jumps, Ann. Mat. Pura Appl., 197, 1311-1332 (2018) · Zbl 1435.34063
[23] M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat., 114 (1989), 187-209. · Zbl 0671.26006
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