A metric approach to a class of doubly nonlinear evolution equations and applications
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 1, pp. 97-169.

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in L 1 spaces.

Classification: 35K55, 49Q20, 58E99
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     title = {A metric approach to a class of doubly nonlinear evolution equations and applications},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Rossi, Riccarda; Mielke, Alexander; Savaré, Giuseppe. A metric approach to a class of doubly nonlinear evolution equations and applications. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 1, pp. 97-169. http://archive.numdam.org/item/ASNSP_2008_5_7_1_97_0/

[1] R. A. Adams, “Sobolev Spaces”, Pure and Applied Mathematics, Academic Press, New York-London, 1975. | MR | Zbl

[2] A. Ambrosetti and G. Prodi, “A Primer of Nonlinear Analysis”, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993. | MR | Zbl

[3] L. Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 439-478. | Numdam | MR | Zbl

[4] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191-246. | MR | Zbl

[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. | MR | Zbl

[6] T. Arai, On the existence of the solution for φ(u ' (t))+ψ(u(t))f(t), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), 75-96. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[9] H. Attouch, “Variational Convergence for Functions and Operators”, Applicable Mathematics Series, Pitman (Advanced Publishing Program) Boston MA, 1984. | MR | Zbl

[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. | MR | Zbl

[11] E. J. Balder, An Extension of Prokhorov's Theorem for Transition Probabilities with Applications to Infinite-Dimensional Lower-Closure Problems, Rend. Circ. Mat. Palermo 34 (1985), 427-447. | MR | Zbl

[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. | MR | Zbl

[13] V. Barbu, Existence theorems for a class of two point boundary problems, J. Differential Equations 17 (1975), 236-257. | MR | Zbl

[14] C. Castaing and M. Valadier, “Convex Analysis and Measurable Multifunctions”, Springer, Berlin-New York, 1977. | MR | Zbl

[15] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), 737-756. | MR | Zbl

[16] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math. 9 (1992), 181-203. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] G. Dal Maso, G. Francfort and R. Toader, Quasistatic growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), 165-225. | MR | Zbl

[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. | MR | Zbl

[23] L. C. Evans and R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton FL, 1992. | MR | Zbl

[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. | MR | Zbl

[25] P. Germain, “Cours de Mécanique des Milieux Continus. Tome I: Théorie Générale”, Masson et Cie Éditeurs, Paris, 1973. | MR | Zbl

[26] M. Giaquinta, “Multiple Integrals in the Calculus of Variations”, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1983. | MR | Zbl

[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. | Numdam | MR | Zbl

[28] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005), 73-99. | MR | Zbl

[29] A. Mielke, Finite elastoplasticity Lie groups and geodesics on SL(d), In: “Geometry, Mechanics, and Dynamics”, Springer, New York, 2002, 61-90. | MR | Zbl

[30] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn. 15 (2003), 351-382. | MR | Zbl

[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. | MR | Zbl

[32] A. Mielke, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal. 36 (2004), 384-404. | MR | Zbl

[33] A. Mielke, Evolution of rate-independent systems, In: “Evolutionary Equations”, Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461-559. | MR | Zbl

[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. | Numdam | MR

[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. | MR | Zbl

[36] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, in preparation. | Zbl

[37] A. Mielke, R. Rossi and G. Savaré, On the vanishing viscosity limit for the metric approach to rate-independent problems, in preparation.

[38] A. Mielke and T. Roubiček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul. 1 (2003), 571-597. | MR | Zbl

[39] A. Mielke and T. Roubiček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci. 16 (2006), 177-209. | MR | Zbl

[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] 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. | MR | Zbl

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

[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. | MR | Zbl

[44] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115-162. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[47] A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst. 14 (2006), 801-820. | MR | Zbl

[48] T. Senba, On some nonlinear evolution equations, Funkcial. Ekva. 29 (1986), 243-257. | MR | Zbl

[49] U. Stefanelli, The Brézis-Ekeland principle for doubly nonlinear equations, to appear in SIAM J. Control Optim. | MR | Zbl

[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. | MR | Zbl

[51] A. Visintin, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 233-238. | MR | Zbl