{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:54:54Z","timestamp":1787320494389,"version":"build-2736575974"},"reference-count":105,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>The celebrated Brezis\u2013Ekeland principle [C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Ai, A1197\u2013A1198, Aii, and A971\u2013A974] characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems. This note extends this characterization to doubly nonlinear evolution equations driven by convex potentials. The characterization is exploited in order to establish approximation results for gradient flows, doubly nonlinear equations, and rate-independent evolutions.<\/jats:p>","DOI":"10.1137\/070684574","type":"journal-article","created":{"date-parts":[[2008,5,30]],"date-time":"2008-05-30T18:15:22Z","timestamp":1212171322000},"page":"1615-1642","source":"Crossref","is-referenced-by-count":59,"title":["The Brezis\u2013Ekeland Principle for Doubly Nonlinear Equations"],"prefix":"10.1137","volume":"47","author":[{"given":"Ulisse","family":"Stefanelli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,30]]},"reference":[{"key":"R1","first-page":"1","volume":"7","author":"Aizicovici S.","year":"1997","journal-title":"Panamer. Math. J."},{"key":"R2","unstructured":"L. Ambrosio, N. Gigli, and G. Savar\u00e9,\n                      Gradient flows in metric spaces and in the space of probability measures\n                      , in Lectures Math. ETH Z\u00fcrich, Birkh\u00e4user-Verlag, Basel, 2005."},{"key":"R3","first-page":"75","volume":"26","author":"Arai T.","year":"1979","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2004.08.041"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771611"},{"key":"R6","unstructured":"E. Asplund,\n                      Topics in the theory of convex functions\n                      , in Theory and Applications of Monotone Operators (Proceedings of the NATO Advanced Study Institute, Venice, 1968), Edizioni \u201cOderisi\u201d, Gubbio, 1969, pp. 1\u201333."},{"key":"R7","unstructured":"H. Attouch,\n                      Variational convergence for functions and operators\n                      , Pitman, Boston, 1984."},{"key":"R8","unstructured":"J.P. Aubin,\n                      Variational principles for differential equations of elliptic, parabolic and hyperbolic type\n                      , in Mathematical Techniques of Optimization, Control and Decision, Birk-h\u00e4user, Boston, 1981, pp. 31\u201345."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"J.P. Aubin and A. Cellina,\n                      Differential Inclusions\n                      , Grundlehren Math. Wiss. 264, Springer-Verlag, Berlin, 1984.","DOI":"10.1007\/978-3-642-69512-4"},{"key":"R10","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.57262\/die\/1370021920","volume":"6","author":"Auchmuty G.","year":"1993","journal-title":"Differential Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0893-4983","issn-type":"print"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0322035"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(75)90043-1"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"V. Barbu,\n                      Nonlinear Semigroups and Differential Equations in Banach Spaces\n                      , P. Noordhoff, Leyden, 1976.","DOI":"10.1007\/978-94-010-1537-0"},{"key":"R14","first-page":"1959","volume":"10","author":"Barbu V.","year":"1995","journal-title":"Control Theory Adv. Tech.","ISSN":"https:\/\/id.crossref.org\/issn\/0911-0704","issn-type":"print"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.97.1463"},{"key":"R16","unstructured":"H. Brezis,\n                      Op\u00e9rateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert\n                      , North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973."},{"key":"R17","first-page":"Ai,","volume":"282","author":"Brezis H.","year":"1976","journal-title":"C. R. Acad. Sci. Paris Ser. A-B"},{"key":"R18","first-page":"Aii,","volume":"282","author":"Brezis H.","year":"1976","journal-title":"C. R. Acad. Sci. Paris Ser. A-B"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"M. Brokate and J. Sprekels,\n                      Hysteresis and Phase Transitions\n                      , Appl. Math. Sci. 121, Springer-Verlag, Berlin, 1996.","DOI":"10.1007\/978-1-4612-4048-8"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"C. Castaing, P. Raynaud de Fitte, and M. Valadier,\n                      Young Measures on Topological Spaces\n                      , Math. Appl. 571, Kluwer Academic, Dordrecht, 2004.","DOI":"10.1007\/1-4020-1964-5"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"C. Castaing and M. Valadier,\n                      Convex analysis and measurable multifunctions\n                      , in Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.","DOI":"10.1007\/BFb0087685"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF03167565"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1080\/03605309908820706"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"G. Dal Maso,\n                      An Introduction to $\\Gamma$-convergence\n                      , Progr. Nonlinear Differential Equations Appl. 8, Birkh\u00e4user Boston, Boston, 1993.","DOI":"10.1007\/978-1-4612-0327-8"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-005-0407-0"},{"key":"R26","unstructured":"G. Dal Maso, A. DeSimone, M. G. Mora, and M. Morini,\n                      A Vanishing Viscosity Approach to Quasistatic Evolution in Plasticity with Softening\n                      , preprint; available online from http:\/\/cvgmt.sns.it\/cgi\/get.cgi\/papers\/daldesmor06\/ (2006)."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-004-0351-4"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"J. Diestel and J. J. Uhl,\n                      Vector Measures\n                      , Math. Surveys Monogr. 15, American Mathematical Society, Providence, RI, 1977.","DOI":"10.1090\/surv\/015"},{"key":"R29","first-page":"151","volume":"13","author":"Efendiev M. A.","year":"2006","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R30","unstructured":"P. Germain,\n                      Cours de M\u00e9canique des Milieux Continus\n                      , Tome I: Th\u00e9orie g\u00e9n\u00e9rale, Masson et Cie, \u00c9diteurs, Paris, 1973."},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.12.008"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.12.010"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2005.4.735"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2006.02.002"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20176"},{"key":"R36","unstructured":"N. Ghoussoub,\n                      Selfdual Partial Differential Systems and Their Variational Principles\n                      , Universitext, Springer-Verlag, to appear."},{"key":"R37","unstructured":"N. Ghoussoub,\n                      Maximal Monotone Operators Are Selfdual Vector Fields and Vice-Versa\n                      , preprint; available online from http:\/\/www.pims.math.ca\/~nassif\/ (2006)."},{"key":"R38","doi-asserted-by":"crossref","unstructured":"N. Ghoussoub and R. J. McCann,\n                      A least action principle for steepest descent in a non-convex landscape\n                      , in Partial Differential Equations and Inverse Problems, Contemp. Math. 362, American Mathematical Society, Providence, RI, 2004, pp. 177\u2013187.","DOI":"10.1090\/conm\/362\/06612"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-004-0558-6"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-006-0021-5"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1007\/s10231-007-0046-1"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2005.12.010"},{"key":"R43","first-page":"842","volume":"58","author":"De Giorgi E.","year":"1975","journal-title":"Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1007\/BF01262691"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1007\/BF00248489"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/99951"},{"key":"R47","unstructured":"W. Han and B. D. Reddy,\n                      Plasticity, Mathematical Theory and Numerical Analysis\n                      , Springer-Verlag, New York, 1999."},{"key":"R48","doi-asserted-by":"crossref","first-page":"278","DOI":"10.21136\/AM.1969.103236","volume":"14","author":"Hlav\u00e1\u010dek I.","year":"1969","journal-title":"Appl. Math."},{"key":"R49","unstructured":"N. Kenmochi and U. Stefanelli,\n                      Existence for a class of nonlocal quasivariational evolution problems\n                      , in Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modelling and Simulation, Chiba 2007, Mathematical Sciences and Applications, Gakuto, to appear."},{"key":"R50","unstructured":"P. Krej\u010d\u00ed,\n                      Hysteresis, Convexity and Dissipation in Hyperbolic Equations\n                      , GAKUTO Internat. Ser. Math. Sci. Appl. 8, Gakkotosho, Tokyo, 1996."},{"key":"R51","first-page":"63","volume":"3","author":"Lemaire B.","year":"1996","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R52","first-page":"47","volume":"318","author":"Mabrouk M.","year":"1994","journal-title":"C. R. Acad. Sci. Paris Ser. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"R53","doi-asserted-by":"publisher","DOI":"10.5802\/afst.955"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(00)01826-7"},{"key":"R55","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(03)00018-6"},{"key":"R56","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-004-0267-8"},{"key":"R57","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-7824(02)01276-X"},{"key":"R58","doi-asserted-by":"crossref","unstructured":"A. Mielke,\n                      Finite Elastoplasticity Lie Groups and Geodesics on ${\\rmSL}(d)$\n                      , in Geometry, Mechanics, and Dynamics, Springer-Verlag, New York, 2002, pp. 61\u201390.","DOI":"10.1007\/0-387-21791-6_2"},{"key":"R59","doi-asserted-by":"publisher","DOI":"10.1007\/s00161-003-0120-x"},{"key":"R60","doi-asserted-by":"crossref","unstructured":"A. Mielke,\n                      Evolution of rate-independent inelasticity with microstructure using relaxation and Young measures\n                      , in IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Stuttgart, 2001), Solid Mech. Appl. 108, Kluwer Academic, Dordrecht, 2003, pp. 33\u201344.","DOI":"10.1007\/978-94-017-0297-3_3"},{"key":"R61","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141003429906"},{"key":"R62","doi-asserted-by":"crossref","unstructured":"A. Mielke,\n                      Evolution of rate-independent systems\n                      , in Handbook of Differential Equations, Evolutionary Equations, Vol. 2, C. Dafermos and E. Feireisl, eds., Elsevier, New York, 2005, pp. 461\u2013559.","DOI":"10.1016\/S1874-5717(06)80009-5"},{"key":"R63","unstructured":"A. Mielke and M. Ortiz,\n                      A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems\n                      , ESAIM Control Optim. Calc. Var., to appear; preprint available online from http:\/\/www.wias-berlin.de\/main\/publications\/wias-publ\/index.cgi.en."},{"key":"R64","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250700184X"},{"key":"R65","unstructured":"A. Mielke, R. Rossi, and G. Savar\u00e9,\n                      Non parametric rate-independent flows\n                      , manuscript, 2008."},{"key":"R66","unstructured":"A. Mielke, R. Rossi, and G. Savar\u00e9,\n                      A metric approach to a class of doubly nonlinear evolution equations and application\n                      , Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear; preprint available online from http:\/\/www.wias-berlin.de\/main\/publications\/wias-publ\/index.cgi.en."},{"key":"R67","doi-asserted-by":"publisher","DOI":"10.1137\/S1540345903422860"},{"key":"R68","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250600111X"},{"key":"R69","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-007-0119-4"},{"key":"R70","doi-asserted-by":"crossref","unstructured":"A. Mielke and U. Stefanelli,\n                      A Discrete Variational Principle for Rate-Independent Evolution\n                      , WIAS preprint 1295, 2008; available online from http:\/\/wias-berlin.de\/main\/publications\/wias-publ\/index.cgi.en.","DOI":"10.1515\/ACV.2008.017"},{"key":"R71","unstructured":"A. Mielke and F. Theil,\n                      A mathematical model for rate-independent phase transformations with hysteresis\n                      , in Proceedings of the Workshop on \u201cModels of Continuum Mechanics in Analysis and Engineering,\u201d H.D. Alber, R. Balean, and R. Farwig, eds., Shaker-Verlag, Aachen, 1999, pp. 117\u2013129."},{"key":"R72","doi-asserted-by":"publisher","DOI":"10.1007\/s002050200194"},{"key":"R73","doi-asserted-by":"publisher","DOI":"10.1002\/mma.731"},{"key":"R74","first-page":"608","volume":"271","author":"Moreau J.-J.","year":"1970","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. II M\u00e9c. Phys. Chim. Sci. Univers Sci. Terre"},{"key":"R75","first-page":"A118","volume":"273","author":"Moreau J.-J.","year":"1971","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"R76","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(69)90009-7"},{"key":"R77","first-page":"Aiv,","volume":"282","author":"Nayroles B.","year":"1976","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"R78","first-page":"22","volume":"6","author":"Nayroles B.","year":"1976","journal-title":"Travaux S\u00e9m. Anal. Convexe"},{"key":"R79","unstructured":"C. Ortner,\n                      Two Variational Techniques for the Approximation of Curves of Maximal Slope\n                      , Technical report NA05\/10, Oxford University Computing Laboratory, 2005."},{"key":"R80","doi-asserted-by":"publisher","DOI":"10.1137\/050643982"},{"key":"R81","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141003425696"},{"key":"R82","doi-asserted-by":"publisher","DOI":"10.1137\/040614578"},{"key":"R83","first-page":"A93","volume":"275","author":"Peralba J.-C.","year":"1972","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"R84","unstructured":"J.C. Peralba,\n                      Un probl\u00e8me d'\u00e9volution relatif \u00e0 un op\u00e9rateur sous-diff\u00e9rentiel d\u00e9pendant du temps\n                      , in Travaux du S\u00e9minaire d'Analyse Convexe, Vol. II, Exp. 6, U.E.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier, 1972, p. 17."},{"key":"R85","first-page":"Ai,","volume":"283","author":"Rios H.","year":"1976","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"R86","first-page":"16","volume":"6","author":"Rios H.","year":"1976","journal-title":"Travaux S\u00e9m. Anal. Convexe"},{"key":"R87","first-page":"96","volume":"8","author":"Rios H.","year":"1978","journal-title":"Travaux S\u00e9m. Anal. Convexe"},{"key":"R88","doi-asserted-by":"publisher","DOI":"10.5802\/afst.538"},{"key":"R89","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2006013"},{"key":"R90","first-page":"57","volume":"10","author":"Roub\u00ed\u010dek T.","year":"2000","journal-title":"Adv. Math. Sci. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1343-4373","issn-type":"print"},{"key":"R91","unstructured":"T. Roub\u00ed\u010dek,\n                      Nonlinear Partial Differential Equations with Applications\n                      , Internat. Ser. Numer. Math. 153, Birkh\u00e4user-Verlag, Basel, 2005."},{"key":"R92","first-page":"137","volume":"33","author":"Salvadori A.","year":"1984","journal-title":"Atti Sem. Mat. Fis. Univ. Modena","ISSN":"https:\/\/id.crossref.org\/issn\/0041-8986","issn-type":"print"},{"key":"R93","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20046"},{"key":"R94","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792505006352"},{"key":"R95","first-page":"243","volume":"29","author":"Senba T.","year":"1986","journal-title":"Funkcial. Ekval."},{"key":"R96","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762360"},{"key":"R97","doi-asserted-by":"crossref","first-page":"897","DOI":"10.57262\/die\/1356060778","volume":"15","author":"Stefanelli U.","year":"2002","journal-title":"Differential Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0893-4983","issn-type":"print"},{"key":"R98","doi-asserted-by":"publisher","DOI":"10.1007\/s00028-003-0056-4"},{"key":"R99","first-page":"319","volume":"7","author":"Stefanelli U.","year":"2004","journal-title":"Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat."},{"key":"R100","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2006.05.004"},{"key":"R101","unstructured":"U. Stefanelli,\n                      The Discrete Brezis-Ekeland Principle\n                      , J. Convex. Anal., to appear."},{"key":"R102","unstructured":"U. Stefanelli,\n                      Some remarks on convergence and approximation for a class of hysteresis problems\n                      , Istit. Lombardo Accad. Sci. Lett. Rend. A; preprint available online http:\/\/www.imati.cnr.it\/ulisse\/pubbl.html."},{"key":"R103","unstructured":"U. Stefanelli,\n                      A variational principle for hardening elasto-plasticity\n                      , SIAM J. Math. Anal., to appear; preprint available from http:\/\/www.imati.cnr.it\/ulisse\/pubbl.html."},{"key":"R104","doi-asserted-by":"crossref","unstructured":"A. Visintin,\n                      Differential Models of Hysteresis\n                      , Appl. Math. Sci. 111, Springer-Verlag, Berlin, 1994.","DOI":"10.1007\/978-3-662-11557-2"},{"key":"R105","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(00)01825-5"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070684574","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:58:24Z","timestamp":1787317104000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070684574"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":105,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070684574"],"URL":"https:\/\/doi.org\/10.1137\/070684574","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}