Quasiconvexity at the boundary, positivity of the second variation and elastic stability

V. I. Arnold [1969]. On an a priori estimate in the theory of hydrodynamic stability, Am. Math. Soc. Transl. 79, 267?269.

J. M. Ball [1977a]. Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337?403.

J. M. Ball [1977b]. Constitutive inequalities and existence theorems in elasticity, in Nonlinear Analysis and Mechanics, Vol. I, R. J. Knops (ed), Pitman.

J. M. Ball [1981]. Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh 88A, 315?328.

J. M. Ball [1982]. Discontinuous equilibrium solutions and cavitation in non-linear elasticity, Phil. Trans. R. Soc. London A 306, 557?611.

J. M. Ball [1984]. Differentiability properties of symmetric and isotropic functions, Duke Math. J., to appear.

J. M. Ball, J. C. Currie & P. J. Olver [1981]. Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Functional Anal. 41, 135?174.

J. M. Ball & F. Murat [1984]. W1,p-quasiconvexity and variational problems for multiple integrals J. Functional Anal., to appear.

J. M. Ball, R. J. Knops & J. E. Marsden [1978]. Two examples in nonlinear elasticity, Springer Lecture Notes in Mathematics, No. 466, 41?49.

O. Bolza [1904]. Lectures on the Calculus of Variations, Reprinted by Chelsea, N.Y., [1973].

R. C. Browne [1978]. Dynamic stability of one dimensional nonlinearly viscoelastic bodies, Arch. Rational Mech. Anal. 68, 257?282.

H. J. Buchner, J. Marsden & S. Schecter [1983]. Examples for the infinite dimensional Morse Lemma, SIAM J. Math. An. 14, 1045?1055.

H. Busemann & G. C. Shephard [1965]. Convexity on nonconvex sets, Proc. Coll. on Convexity, Copenhagen, Univ. Math. Inst., Copenhagen, 20?33.

P. G. Ciarlet & G. Geymonat [1982]. Sur les lois de comportement en élasticité nonlinéaire compressible, C. R. Acad. Sc. Paris, 295, 423?426.

R. J. DiPerna [1983]. Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82, 27?70.

J. L. Ericksen [1966a]. Thermoelastic stability, Proc. Fifth U.S. Cong. on Appl. Mech. 187?193.

J. L. Ericksen [1966b]. A thermokinetic view of elastic stability theory, Int. Journal Solids Structures, 2, 573?580.

J. L. Ericksen [1975]. Equilibrium of bars, J. of Elasticity 5, 191?201.

J. L. Ericksen [1980]. Some phase transitions in crystals, Arch. Rational Mech. Anal. 73, 99?124.

M. Golubitsky & J. Marsden [1982]. The Morse Lemma in infinite dimensions via singularity theory, SIAM J. Math. An. 14, 1037?1044.

L. M. Graves [1939]. The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5, 656?660.

M. E. Gurtin [1975]. Thermodynamics and stability, Arch. Rational Mech. Anal. 59, 63?96.

M. E. Gurtin [1983]. Two-phase deformations of elastic solids, (preprint).

J. Hadamard [1902]. Sur une question de calcul des variations, Bull. Soc. Math. France 30, 253?256.

P. Hartman [1964]. Ordinary Differential Equations. New York: John Wiley & Sons, Inc., reprinted by Birkhauser, Boston, 1982.

M. R. Hestenes [1966]. Calculus of variations and optimal control theory, Wiley.

D. D. Holm, J. E. Marsden, T. Ratiu & A. Weinstein [1983]. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Physics Letters 98A, 15?21.

T. Hughes, T. Kato & J. Marsden [1977]. Well-posed quasi-linear hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273?294.

R. D. James [1979]. Co-existent phases in the one-dimensional static theory of elastic bars, Arch. Rational Mech. Anal. 72, 99?140.

R. D. James [1980]. The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125?158.

R. D. James [1981]. Finite deformation by mechanical twinning, Arch. Rational Mech. Anal. 77, 143?176.

W. T. Koiter [1945]. On the stability of elastic equilibrium, Dissertation. Delft, Holland (English translation: NASA Tech. Trans. F10, 833 (1967)).

W. T. Koiter [1976]. A basic open problem in the theory of elastic stability, Springer Lecture Notes in Math. 503, 366?373.

W. T. Koiter [1981]. Elastic stability, buckling and post-buckling behaviour, in Proc. IUTAM Symp. on Finite Elasticity, pp. 13?24, D. E. Carlson and R. T. Shield, (eds.), Martinus Nijhoff Publishers.

R. J. Knops & L. E. Payne [1978]. On potential wells and stability in nonlinear elasticity, Math. Proc. Camb. Phil. Soc. 84, 177?190.

R. J. Knops & E. W. Wilkes [1973]. Theory of elastic stability, in Handbuch der Physik VIa/3, C. Truesdell, ed., Springer.

J. E. Marsden & T. J. R. Hughes [1983]. Mathematical Foundations of Elasticity, Prentice-Hall.

R. Martini [1979]. On the Fréchet differentiability of certain energy functionals, Proc. Kon Ned. Akad. Wet. B 82: 42?45.

N. G. Meyers [1965]. Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc. 119, 225?249.

C. B. Morrey [1952]. Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 25?53.

C. B. Morrey [1966]. Multiple Integrals in the Calculus of Variations, Springer.

M. Potier-Ferry [1982]. On the mathematical foundations of elastic stability theory. I., Arch. Rational Mech. Anal. 78, 55?72.

H. Rund [1963]. On the Weierstrass excess function of parameter-invariant multiple integrals in the calculus of variations. Tydskr. Natuurwetensk 3, 168?179.

H. Rund [1974]. Integral formulae associated with the Euler-Lagrange operators of multiple integral problems in the Calculus of Variations, Aeq. Math. 11, 212?229.

D. H. Sattinger [1969]. On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 148?172.

R. T. Shield & A. E. Green [1963]. On certain methods in the stability theory of continuous systems, Arch. Rational Mech. Anal. 12, 354?360.

T. Valent [1981]. Local theorems of existence and uniqueness in finite elastostatics, in Proc. IUTAM Symp. on Finite Elasticity, pp. 401?421, D. T. Carlson & R. T. Shield (eds.), Martinus Nijhoff Publishers.

L. van Hove [1949]. Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues, Koninkl. Belg. Acad., Klasse der Wetenschappen, Verhandelingen, Vol. 24.

Y. H. Wan, J. E. Marsden, T. Ratiu & A. Weinstein [1983]. Nonlinear stability of circular vortex patches (to appear).