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Ordinary differential equations of non-linear elasticity. II: Existence and regularity theory for conservative boundary value problems. (English) Zbl 0354.73047


MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K25 Shells
74K15 Membranes
47E05 General theory of ordinary differential operators
49J15 Existence theories for optimal control problems involving ordinary differential equations
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[1] N.I. Akhiezer (1955), Lectures on the Calculus of Variations (in Russian). Gostekhteorizdat, Moscow. (English translation by A.H. Frink (1962). Blaisdell, New York).
[2] A. Ambrosetti & P.H. Rabinowitz (1973), Dual variational methods in critical point theory and applications. J. Functional Analysis 14, 349–381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[3] S.S. Antman (1968), Equilibrium states of nonlinearly elastic rods. J. Math. Anal. Appl. 23, 459–470. · Zbl 0165.27704 · doi:10.1016/0022-247X(68)90083-8
[4] S.S. Antman (1970a), The shape of buckled nonlinearly elastic rings. Z. Angew. Math. Phys. 21, 422–438. · Zbl 0214.52005 · doi:10.1007/BF01627947
[5] S.S. Antman (1970b), Existence of solutions of the equilibrium equations for nonlinearly elastic rings and arches. Indiana University Math. J. 20, 281–302. · Zbl 0223.73029 · doi:10.1512/iumj.1971.20.20025
[6] S.S. Antman (1971), Existence and uniqueness of axisymmetric equilibrium states of nonlinearly elastic shells. Arch. Rational Mech. Anal. 40, 329–372. · Zbl 0254.73072 · doi:10.1007/BF00251796
[7] S.S. Antman (1972), The Theory of Rods. Handbuch der Physik VIa/2, edited by C. Truesdell. Springer-Verlag, Berlin, Heidelberg, New York, 641–703.
[8] S.S. Antman (1973a), Nonuniqueness of equilibrium states for bars in tension. J. Math. Anal. Appl. 44, 333–349. · Zbl 0267.73031 · doi:10.1016/0022-247X(73)90063-2
[9] S. Antman (1973b), Monotonicity and invertibility conditions in one-dimensional nonlinear elasticity, Symposium on Nonlinear Elasticity, edited by R.W. Dickey. Academic Press, New York, 57–92.
[10] S.S. Antman (1974), Qualitative theory of the ordinary differential equations of nonlinear elasticity, Mechanics Today, 1972, edited by S. Nemat-Nasser. Pergamon Press, New York, 58–101.
[11] S.S. Antman & E. Carbone (1976), Shear and necking instabilities in nonlinear elasticity. J. Elasticity, to appear. · Zbl 0356.73048
[12] S.S. Antman & K.B. Jordan (1975), Qualitative aspects of the spatial deformation of nonlinearly elastic rods. Proc. Roy. Soc. Edinburgh, 73A, 85–105. · Zbl 0351.73076 · doi:10.1017/S0308210500016309
[13] S.S. Antman & G. Rosenfeld (1976), Global behavior of buckled states of nonlinearly elastic rods, to appear. · Zbl 0395.73039
[14] J.F. Bell (1973), The Experimental Foundations of Solid Mechanics. Handbuch der Physik VIa/1, edited by C. Truesdell. Springer-Verlag, Berlin, Heidelberg, New York.
[15] F.E. Browder (1965a), Variational methods for nonlinear elliptic eigenvalue problems. Bull. Am. Math. Soc. 71, 176–183. · Zbl 0135.15802 · doi:10.1090/S0002-9904-1965-11275-7
[16] F.E. Browder (1965b), Remarks on the direct method of the calculus of variations. Arch. Rational Mech. Anal. 20, 251–258. · Zbl 0135.32404 · doi:10.1007/BF00253135
[17] M.D. Cannon, D.D. Cullum & E. Polak (1970), Theory of Optimal Control and Mathematical Programming. McGraw-Hill, New York.
[18] L. Cesari (1974), Lower semicontinuity and lower closure theorems without seminormality conditions. Ann. Mat. Pura Appl. 98, 381–397. · Zbl 0281.49006 · doi:10.1007/BF02414036
[19] E.A. Coddington & N. Levinson (1955), Theory of Ordinary Differential Equations. McGraw-Hill, New York. · Zbl 0064.33002
[20] N. Dunford & J.T. Schwartz (1958), Linear Operators, Part I. Interscience, New York.
[21] G. Fichera (1964), Semicontinuity of multiple integrals in ordinary forms. Arch. Rational Mech. Anal. 16, 339–352. · Zbl 0128.10003 · doi:10.1007/BF00250470
[22] J.M. Greenberg (1967), On the equilibrium configurations of compressible slender bars. Arch. Rational Mech. Anal. 27, 181–194. · Zbl 0162.56404 · doi:10.1007/BF00290612
[23] J.K. Hale (1969), Ordinary Differential Equations. Wiley-Interscience, New York. · Zbl 0186.40901
[24] M. Hestenes (1966), Calculus of Variations and Optimal Control Theory. Wiley, New York. · Zbl 0173.35703
[25] V.I. Kazimirov (1956), On the semi-continuity of integrals in the calculus of variations (in Russian). Usp. Mat. Nauk 11, No. 3 (69), 125–129.
[26] M.A. Krasnosel’skii & Ya. B. Rutitskii (1958), Convex Functions and Orlicz Spaces (in Russian). Fizmatgiz, Moscow. (English translation by L. F. Boron (1961), Noordhoff, Groningen).
[27] C.B. Morrey, Jr. (1966), Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York. · Zbl 0142.38701
[28] J. Necas (1967), Les méthodes directes en théorie des équations elliptiques. Masson, Paris.
[29] F. Odeh & I. Tadjbakhsh (1965), A nonlinear eigenvalue problem for rotating rods. Arch. Rational Mech. Anal. 20, 81–94. · Zbl 0136.22304 · doi:10.1007/BF00284611
[30] W.E. Olmstead & D.J. Mescheloff (1974), Buckling of a nonlinear elastic rod. J. Math. Anal. Appl. 45, 609–634. · Zbl 0308.73023 · doi:10.1016/0022-247X(74)90265-0
[31] W.T. Reid (1971), Ordinary Differential Equations. Wiley, New York. · Zbl 0212.10901
[32] R.T. Rockafellar (1970), Convex Analysis. Princeton Univ. Press, Princeton. · Zbl 0193.18401
[33] H. Royden (1968), Real Analysis. Mac Millan, New York. · Zbl 0197.03501
[34] I. Stakgold (1971), Branching of solutions of nonlinear equations. S.I.A.M. Rev. 13, 289–332. · Zbl 0199.20503
[35] J.B. Serrin (1961), On the definition and properties of certain variational integrals. Trans. Am. Math. Soc. 101, 139–167. · Zbl 0102.04601 · doi:10.1090/S0002-9947-1961-0138018-9
[36] C. Truesdell & W. Noll (1965), The Non-Linear Field Theories of Mechanics, Handbuch der Physik III/3, edited by S. Flügge. Springer-Verlag, Berlin, Heidelberg, New York.
[37] C. Truesdell & R. Toupin (1960), The Classical Field Theories. Handbuch der Physik III/1, edited by S. Flügge. Springer-Verlag, Berlin, Heidelberg, New York, 226–793.
[38] M.M. Vainberg (1956), Variational Methods for the Study of Non-Linear Operators, Gostekhteorizdat, Moscow. (English translation by A. Feinstein (1964), Holden-Day, San Francisco.)
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