The Brezis-Ekeland principle for doubly nonlinear equations

SIAM Journal on Control and Optimization

Volumn 47, Issue 3, 2008, Pages 1615-1642

  Stefanelli, Ulisse ^a  

^a CNR   (Italy)

Author keywords

Approximation; Brezis Ekeland principle; Doubly nonlinear equation; Gradient flow; Mosco convergence; Rate independent evolution; Young measures

Indexed keywords

DIFFERENTIAL EQUATIONS; POLYNOMIAL APPROXIMATION;

APPROXIMATION; BREZIS-EKELAND PRINCIPLE; DOUBLY NONLINEAR EQUATION; GRADIENT FLOW; MOSCO CONVERGENCE; RATE-INDEPENDENT EVOLUTION; YOUNG MEASURES;

NONLINEAR EQUATIONS;

EID: 55549128580     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/070684574     Document Type: Article

References (105)

1. S. AIZICOVICI AND Q. YAN, Convergence theorems for abstract doubly nonlinear differential equations, Panamer. Math. J., 7 (1997). pp. 1-17.
2. L. AMBROSIO, N. GIGLI, AND G. SAVARÉ, Gradient flows in metric spaces and in the space of probability measures, in Lectures Math. ETH Zürich, Birkhäuser-Verlag, Basel, 2005.
3. T. ARAI, On the existence of the solution for ∂ξ(u′(t)) + ∂(u(t)) ∋ f(t), J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26 (1979), pp. 75-96.
4. M. ASO, M. FRÉMOND, AND N. KENMOCHI, Phase change problems with temperature dependent constraints for the volume fraction velocities, Nonlinear Anal., 60 (2005), pp. 1003-1023.
5. E. ASPLUND, Averaged norms, Israel J. Math., 5 (1967), pp. 227-233.
6. E. ASPLUND, Topics in the theory of convex functions, in Theory and Applications of Monotone Operators (Proceedings of the NATO Advanced Study Institute, Venice, 196S), Edizioni "Oderisi", Gubbio, 1969, pp. 1-33.
7. H. ATTOUCH, Variational convergence for functions and operators, Pitman, Boston, 1984.
8. J.-P. AUBIN, Variational principles for differential equations of elliptic, parabolic and hyperbolic type, in Mathematical Techniques of Optimization, Control and Decision, Birkhäuser, Boston, 1981, pp. 31-45.
9. J.-P. AUBIN AND A. CELLINA, Differential Inclusions, Grundlehren Math. Wiss. 264, Springer-Verlag, Berlin, 1984.
10. G. AUCHMUTY, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), pp. 1161-1171.
11. E. J. BALDER, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), pp. 570-598.
12. V. BARBU, Existence theorems for a class of two point boundary problems, J. Differential Equations, 17 (1975), pp. 236-257.
13. V. BARBU, Nonlinear Semigroups and Differential Equations in Banach Spaces, P. Noordhoff, Leyden, 1976.
14. V. BARBU AND K. KUNISCH, Identification of nonlinear parabolic equations, Control Theory Adv. Tech., 10 (1995), pp. 1959-1980.
15. M. A. BIOT, Variational principles in irreversible thermodynamics with application to viscoelasticity, Phys. Rev., 97 (1955), pp. 1463-1469.
16. H. BREZIS, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.
17. H. BREZIS AND I. EKELAND, Un principe variationnel associé à certaines équations paraboliques. Le cas dépendant du temps, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Ai, A1197-A1198.
18. H. BREZIS AND I. EKELAND, Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Aii, A971-A974.
19. M. BROKATE AND J. SPREKELS, Hysteresis and Phase Transitions, Appl. Math. Sci. 121, Springer-Verlag, Berlin, 1996.
20. C. CASTAING, P. RAYNAUD DE FITTE, AND M. VALADIER, Young Measures on Topological Spaces, Math. Appl. 571, Kluwer Academic, Dordrecht, 2004.
21. C. CASTAING AND M. VALADIER, Convex analysis and measurable multifunctions, in Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.
22. P. COLLI, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math., 9 (1992), pp. 181-203.
23. P. COLLI AND A. VISINTIN, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), pp. 737-756.
24. G. DAL MASO, An Introduction to Γ- convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser Boston, Boston, 1993.
25. G. DAL MASO, A. DESIMONE, AND M. G. MORA, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), pp. 237-291.
26. G. DAL MASO, A. DESIMONE, M. G. MORA, AND M. MORINI, A Vanishing Viscosity Approach to Quasistatic Evolution in Plasticity with Softening, preprint; available online from http://cvgmt.sns.it/cgi/get.cgi/papers/daldesmor06/(2006).
27. G. DAL MASO, G. A. FRANCFORT, AND R. TOADER, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), pp. 165-225.
28. J. DIESTEL AND J. J. UHL, Vector Measures, Math. Surveys Monogr. 15, American Mathematical Society, Providence, RI, 1977.
29. M. A. EFENDIEV AND A. MIELKE, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), pp. 151-167.
30. P. GERMAIN, Cours de Mécanique des Milieux Continus, Tome I: Théorie générale, Masson et Cie, Éditeurs, Paris, 1973.
31. N. GHOUSSOUB, A theory of anti-selfdual Lagrangians: Dynamical case, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 325-330.
32. N. GHOUSSOUB, A theory of anti-selfdual Lagrangians: Stationary case, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 245-250.
33. N. GHOUSSOUB, A variational principle for nonlinear transport equations, Commun. Pure Appl. Anal., 4 (2005), pp. 735-742.
34. N. GHOUSSOUB, Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), pp. 171-205.
35. N. GHOUSSOUB, Antisymmetric Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions, Comm. Pure Appl. Math., 60 (2007), pp. 619-653.
36. N. GHOUSSOUB, Selfdual Partial Differential Systems and Their Variational Principles, Universitext, Springer-Verlag, to appear.
37. N. GHOUSSOUB, Maximal Monotone Operators Are Selfdual Vector Fields and Vice-Versa, preprint; available online from http://www.pims.math. ca/~nassif/(2006).
38. N. GHOUSSOUB AND R. J. MCCANN, A least action principle for steepest descent in a nonconvex landscape, in Partial Differential Equations and Inverse Problems, Contemp. Math. 362, American Mathematical Society, Providence, RI, 2004, pp. 177-187.
39. N. GHOUSSOUB AND L. TZOU, A variational principle for gradient flows, Math. Ann., 330 (2004), pp. 519-549.
40. N. GHOUSSOUB AND L. TZOU, Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows, Calc. Var. Partial Differential Equations, 26 (2006), pp. 511-534.
41. N. GHOUSSOUB AND L. TZOU, Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations, Ann. Mat. Pura Appl. (4), 187 (2008), pp. 323-352.
42. G. GILARDI AND U. STEFANELLI, Time-discretization and global solution for a doubly nonlinear Volterra equation, J. Differential Equations, 228 (2006), pp. 707-736.
43. E. DE GIORGI AND T. FRANZONI, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), pp. 842-850.
44. M. E. GURTIN, Variational principles in the linear theory of viscoelasticity, Arch. Ration. Mech. Anal., 13 (1963), pp. 179-191.
45. M. E. GURTIN, Variational principles for linear elastodynamics, Arch. Ration. Mech. Anal., 16 (1964), pp. 34-50.
46. M. E. GURTIN, Variational principles for linear initial value problems, Quart. Appl. Math., 22 (1964), pp. 252-256.
47. W. HAN AND B. D. REDDY, Plasticity, Mathematical Theory and Numerical Analysis, Springer-Verlag, New York, 1999.
48. I. HLAVÁČEK, Variational principles for parabolic equations, Appl. Math., 14 (1969), pp. 278-297.
49. N. KENMOCHI AND U. STEFANELLI, Existence for a class of nonlocal quasivariational evolution problems, in Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modelling and Simulation, Chiba 2007, Mathematical Sciences and Applications, Gakuto, to appear.
50. P. KREJČÍ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Internat. Ser. Math. Sci. Appl. 8, Gakkotosho, Tokyo, 1996.
51. B. LEMAIRE, An asymptotical variational principle associated with the steepest descent method for a convex function, J. Convex Anal., 3 (1996), pp. 63-70.
52. M. MABROUK, Sur un problème d'évolution à données mesures; approche variationnelle, C. R. Acad. Sci. Paris Ser. I Math., 318 (1994), pp. 47-52.
53. M. MABROUK, A variational approach for a semi-linear parabolic equation with measure data, Ann. Fac. Sci. Toulouse Math., 9 (2000), pp. 91-112.
54. M. MABROUK, Un principe variationnel pour une équation non linéaire du second ordre en temps, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 381-386.
55. M. MABROUK, A variational principle for a nonlinear differential equation of second order, Adv. in Appl. Math., 31 (2003), pp. 388-419.
56. A. MAINIK AND A. MIELKE, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), pp. 73-99.
57. G. MICHAILLE AND M. VALADIER, Young measures generated by a class of integrands: A narrow epicontinuity and applications to homogenization, J. Math. Pures Appl., 81 (2002), pp. 1277-1312.
58. A. MIELKE, Finite Elastoplasticity Lie Groups and Geodesies on SL(d), in Geometry, Mechanics, and Dynamics, Springer-Verlag, New York, 2002, pp. 61-90.
59. A. MIELKE, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), pp. 351-382.
60. A. MIELKE, Evolution of rate-independent inelasticity with micro structure using relaxation and Young measures, in IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Stuttgart, 2001), Solid Mech. Appl. 108, Kluwer Academic, Dordrecht, 2003, pp. 33-44.
61. A. MIELKE, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36 (2004), pp. 384-404.
62. A. MIELKE, Evolution of rate-independent systems, in Handbook of Differential Equations, Evolutionary Equations, Vol. 2, C. Dafermos and E. Feireisl, eds., Elsevier, New York, 2005, pp. 461-559.
63. A. MIELKE AND M. ORTIZ, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems, ESAIM Control Optim. Calc. Var., to appear; preprint available online from http://www.wias-berlin.de/main/publications/wiaspubl/index.cgi.en.
64. A. MIELKE AND R. Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, Math. Model Methods Appl. Sci., 17 (2007), pp. 81-123.
65. A. MIELKE, R. ROSSI, AND G. SAVARÉ, Non parametric rate-independent flows, manuscript, 2008.
66. A. MIELKE, R. ROSSI, AND G. SAVARÉ, A metric approach to a class of doubly nonlinear evolution equations and application, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear; preprint available online from http://www.wias-berlin.de/main/ publications/wiaspubl/index.cgi.en.
67. A. MIELKE AND T. ROUBÍČEK, A rate-independent model for inelastic behavior of shapememory alloys, Multiscale Model. Simul., 1 (2003), pp. 571-597.
68. A. MIELKE AND T. ROUBÍČEK, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), pp. 177-209.
69. A. MIELKE, T. ROUBÍČEK, AND U. STEFANELLI, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), pp. 125-164.
70. A. MIELKE AND U. STEFANELLI, A Discrete Variational Principle for Rate-Independent Evolution, WIAS preprint 1295, 2008; available online from http://wias-berlin.de/main/publications/wias-publ/index. cgi.en.
71. 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. Earwig, eds., Shaker-Verlag, Aachen, 1999, pp. 117-129.
72. A. MIELKE, F. THEIL, AND V. I. LEVITAS, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), pp. 137-177.
73. A. MIELKE AND A. M. TIMOFTE, An energetic material model for time-dependent ferroelectric behavior: Existence and uniqueness, Math. Models Appl. Sci., 29 (2005), pp. 1393-1410.
74. J.-J. MOREAU, Sur les lois de frottement, de viscosité et plasticité, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 271 (1970), pp. 608-611.
75. J.-J. MOREAU, Sur l'évolution d'un système élasto-visco-plastique, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), pp. A118-A121.
76. U. MOSCO, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), pp. 510-585.
77. B. NAYROLES, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), pp. Aiv, A1035-A1038.
78. B. NAYROLES, Un théorème de minimum pour certains systèmes dissipatifs. Variante hilbertienne, Travaux Sém. Anal. Convexe, 6 (1976), p. 22.
79. C. ORTNER, Two Variational Techniques for the Approximation of Curves of Maximal Slope, Technical report NA05/10, Oxford University Computing Laboratory, 2005.
80. C. ORTNER., Gradient flows as a selection procedure for equilibria of non-convex energies, SIAM J. Math. Anal., 38 (2006), pp. 1214-1234.
81. P. PEDREGAL, Γ-convergence through Young measures, SIAM J. Math. Anal., 36 (2004), pp. 423-440.
82. P. PEDREGAL, Young measures associated with homogenization, SIAM J. Math. Anal., 37 (2006), pp. 1454-1464.
83. J.-C. PERALBA, Un problème d'évolution relatif à un opérateur sous-différentiel dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 275 (1972), pp. A93-A96.
84. J.-C. PERALBA, Un problème d'évolution relatif à un opérateur sous-différentiel dépendant du temps, in Travaux du Séminaire d'Analyse Convexe, Vol. II, Exp. 6, U.E.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier, 1972, p. 17.
85. H. RIOS, Étude de la question d'existence pour certains problèmes d'évolution par minimisation d'une fonctionnelle convexe, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), pp. Ai, A83-A86.
86. H. RIOS, La question d'existence de solutions pour certaines équations à opérateurs monotones vue comme problème de minimum ou comme problème de point-selle, Travaux Sém. Anal. Convexe, 6 (1976), p. 16.
87. H. RIOS, Étude de certains problèmes paraboliques: Existence et approximation des solutions, Travaux Sém. Anal. Convexe, 8 (1978), p. 96.
88. H. RIOS, Une étude d'existence sur certains problèmes paraboliques, Ann. Fac. Sci. Toulouse Math. (5), 1 (1979), pp. 235-255.
89. G. ROSSI AND G. SAVARÉ, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 564-614.
90. T. ROUBÍČEK, Direct method for parabolic problems, Adv. Math. Sci. Appl., 10 (2000), pp. 57-65.
91. T. ROUBÍČEK, Nonlinear Partial Differential Equations with Applications, Internat. Ser. Numer. Math. 153, Birkhäuser-Verlag, Basel, 2005.
92. X Atti Sem. Mat. Fis. Univ. Modena, 33 (1984), pp. 137-154.
93. E. SANDIER AND S. SERFATY, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), pp. 1627-1672.
94. F. SCHMID AND A. MIELKE, Vortex pinning in super-conductivity as a rate-independent process, European J. Appl. Math., 16 (2005), pp. 799-808.
95. T. SENBA, On some nonlinear evolution equation, Funkcial. Ekval., 29 (1986), pp. 243-257.
96. p(0, T; B), Ann. Mat. Pura Appl., 146 (1987), pp. 65-96.
97. U. STEFANELLI, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), pp. 897-922.
98. U. STEFANELLI, On some nonlocal evolution equations in Banach spaces, J. Evol. Equ., 4 (2004), pp. 1-26.
99. U. STEFANELLI, Some quasivariational problems with memory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), pp. 319-333.
100. U. STEFANELLI, Nonlocal quasivariational evolution problems, J. Differential Equations, 229 (2006), pp. 204-228.
101. U. STEFANELLI, The Discrete Brezis-Ekeland Principle, J. Convex. Anal., to appear.
102. U. STEFANELLI, Some remarks on convergence and approximation for a class of hysteresis problems, Istit. Lombardo Accad. Sci. Lett. Rend. A; preprint available online http://www.imati.cnr.it/ulisse/pubbl.html.
103. U. STEFANELLI, A variational principle for hardening elasto-plasticity, SIAM J. Math. Anal., to appear; preprint available from http://www.imati.cnr.it/ulisse/pubbl.html.
104. A. VISINTIN, Differential Models of Hysteresis, Appl. Math. Sci. 111, Springer-Verlag, Berlin, 1994.
105. A. VISINTIN, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), pp. 233-238.