A metric approach to a class of doubly nonlinear evolution equations and applications

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1. [1] R. A. Adams, “Sobolev Spaces”, Pure and Applied Mathematics, Academic Press, New York-London, 1975. Zbl1098.46001MR450957
2. [2] A. Ambrosetti and G. Prodi, “A Primer of Nonlinear Analysis”, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993. Zbl0818.47059MR1225101
3. [3] L. Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 439–478. Zbl0724.49027MR1079985
4. [4] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191–246. Zbl0957.49029MR1387558
5. [5] L. Ambrosio, N. Gigli and G. Savaré, “Gradient Flows in Metric Spaces and in the Wasserstein Spaces of Probability Measures”, Lecture notes, ETH, Birkhäuser, 2005. Zbl1090.35002MR2129498
6. [6] T. Arai, On the existence of the solution for $\partial \phi \left(u^{\text{'}}\left(t\right)\right)+\partial \psi \left(u\left(t\right)\right)\ni f\left(t\right)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math.26 (1979), 75–96. Zbl0418.35056MR539774
7. [7] M. Aso, M. Frémond and N. Kenmochi, Quasi-variational evolution problems for irreversible phase change, In: “Nonlinear Partial Differential Equations and their Applications”, GAKUTO Internat. Ser. Math. Appl., Gakkōtosho, Tokyo, 2004, 517–535. Zbl1061.35032MR2087495
8. [8] M. Aso, M. Frémond and N. Kenmochi, Phase change problems with temperature-dependent constraints for the volume fraction velocities, Nonlinear Anal.60 (2005), 1003–1023. Zbl1058.35085MR2115030
9. [9] H. Attouch, “Variational Convergence for Functions and Operators”, Applicable Mathematics Series, Pitman (Advanced Publishing Program) Boston MA, 1984. Zbl0561.49012MR773850
10. [10] E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim.22 (1984), 570–598. Zbl0549.49005MR747970
11. [11] E. J. Balder, An Extension of Prokhorov’s Theorem for Transition Probabilities with Applications to Infinite-Dimensional Lower-Closure Problems, Rend. Circ. Mat. Palermo34 (1985), 427–447. Zbl0606.60006MR848120
12. [12] J. M. Ball, A version of the fundamental theorem for Young measures, In: “PDEs and Continuum Models of Phase Transitions” (Nice 1988), Lecture Notes in Phys., Vol. 344, Springer, Berlin, 1989, 207–215. Zbl0991.49500MR1036070
13. [13] V. Barbu, Existence theorems for a class of two point boundary problems, J. Differential Equations17 (1975), 236–257. Zbl0295.35074MR380532
14. [14] C. Castaing and M. Valadier, “Convex Analysis and Measurable Multifunctions”, Springer, Berlin-New York, 1977. Zbl0346.46038MR467310
15. [15] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations15 (1990), 737–756. Zbl0707.34053MR1070845
16. [16] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math.9 (1992), 181–203. Zbl0757.34051MR1170721
17. [17] E. De Giorgi, New problems on minimizing movements, In: “Boundary Value Problems for PDE and Applications”, Claudio Baiocchi and Jacques Louis Lions (eds.), Masson, Paris, 1993, 81–98. Zbl0851.35052MR1260440
18. [18] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68 (1980), 180–187. Zbl0465.47041MR636814
19. [19] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal.180 (2006), 237–291. Zbl1093.74007MR2210910
20. [20] G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, to appear in Arch. Ration. Mech. Anal. Zbl1219.35305MR2424994
21. [21] G. Dal Maso, G. Francfort and R. Toader, Quasistatic growth in nonlinear elasticity, Arch. Ration. Mech. Anal.176 (2005), 165–225. Zbl1064.74150MR2186036
22. [22] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal.13 (2006), 151–167. Zbl1109.74040MR2211809
23. [23] L. C. Evans and R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton FL, 1992. Zbl0804.28001MR1158660
24. [24] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math.595 (2006), 55–91. Zbl1101.74015MR2244798
25. [25] P. Germain, “Cours de Mécanique des Milieux Continus. Tome I: Théorie Générale”, Masson et Cie Éditeurs, Paris, 1973. Zbl0254.73001MR368541
26. [26] M. Giaquinta, “Multiple Integrals in the Calculus of Variations”, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1983. Zbl0516.49003MR717034
27. [27] A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 281–330. Zbl0699.49015MR1041899
28. [28] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations22 (2005), 73–99. Zbl1161.74387MR2105969
29. [29] A. Mielke, Finite elastoplasticity Lie groups and geodesics on $SL\left(d\right)$, In: “Geometry, Mechanics, and Dynamics”, Springer, New York, 2002, 61–90. Zbl1146.74309MR1919826
30. [30] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn.15 (2003), 351–382. Zbl1068.74522MR1999280
31. [31] A. Mielke, Evolution of rate-independent inelasticity with microstructure using relaxation and Young measures, In: “IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains” (Stuttgart, 2001), Solid Mech. Appl. 108, Kluwer Acad. Publ., Dordrecht, 2003, 33–44. Zbl1040.74011MR1991322
32. [32] A. Mielke, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal.36 (2004), 384–404. Zbl1076.74012MR2111782
33. [33] A. Mielke, Evolution of rate-independent systems, In: “Evolutionary Equations”, Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461–559. Zbl1120.47062MR2182832
34. [34] A. Mielke and M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems, to appear in ESAIM Control Optim. Calc. Var., published online: 21 December 2007, DOI: 10.1051/cocv: 2007064. Zbl05309728MR2434063
35. [35] 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. Zbl1121.34052MR2290410
36. [36] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, in preparation. Zbl1170.49036
37. [37] A. Mielke, R. Rossi and G. Savaré, On the vanishing viscosity limit for the metric approach to rate-independent problems, in preparation. Zbl1170.49036
38. [38] A. Mielke and T. Roubiček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul.1 (2003), 571–597. Zbl1183.74207MR2029592
39. [39] A. Mielke and T. Roubiček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci.16 (2006), 177–209. Zbl1094.35068MR2210087
40. [40] A. Mielke and F. Theil, A mathematical model for rate–independent phase transformations with hysteresis, In: Proceedings of the Workshop on “Models of Continuum Mechanics in Analysis and Engineering”, H.-D. Alber, R. Balean and R. Farwig (eds.), Shaker–Verlag, Aachen, 1999, 117–129. 
41. [41] A. Mielke, F. Theil and V. Levitas, A variational formulation of rate–independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal.162 (2002), 137–177. Zbl1012.74054MR1897379
42. [42] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl.11 (2004), 151–189. Zbl1061.35182MR2210284
43. [43] J. J. Moreau, Sur l’évolution d’un système élasto-visto-plastique, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A118–A121. Zbl0245.73029MR284066
44. [44] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. Zbl0088.07601MR109940
45. [45] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var.12 (2006), 564–614. Zbl1116.34048MR2224826
46. [46] G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst.18 (2007), 15–38. Zbl1195.35185MR2276484
47. [47] A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst.14 (2006), 801–820. Zbl1092.37052MR2177098
48. [48] T. Senba, On some nonlinear evolution equations, Funkcial. Ekva.29 (1986), 243–257. Zbl0627.35045MR904541
49. [49] U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, to appear in SIAM J. Control Optim. Zbl1194.35214MR2425653
50. [50] M. Valadier, A course on Young measures, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1993), Rend. Istit. Mat. Univ. Trieste 26 (1994), suppl. (1995), 349–394. Zbl0880.49013MR1408956
51. [51] A. Visintin, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math.332 (2001), 233–238. Zbl0977.35142MR1817368

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