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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 87, Issue 3
  • May 1980, pp. 501-513

Strict convexity, strong ellipticity, and regularity in the calculus of variations

  • J. M. Ball (a1)
  • DOI:
  • Published online: 24 October 2008

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

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(1)S. S. Antman and J. E. Osborn The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69 (1979), 231262.

(2)J. M. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337403.

(4)J. M. Ball On the calculus of variations and sequentially weakly continuous maps. In Ordinary and partial differential equations Dundee, 1976, Springer Lecture Notes in Mathematics, vol. 564, 1325.

(7)J. L. Ericksen Equilibrium of bars. J. of Elasticity 5 (1975), 191201.

(12)P. Hartman and C. Olech On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104 (1962), 154178.

(13)J. K. Knowles and E. Sternberg On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elasticity 5 (1975), 341362.

(14)J. K. Knowles and E. Sternberg On the failure of ellipticity of the equations for finite elastic plane strain. Arch. Rational Mech. Anal. 63 (1977), 321326.

(18)R. S. Palais Critical point theory and the minimax principle. Proc. Symp. Pure Math. 15, Amer. Math. Soc. Providence, R. I. (1970), 185212.

(19)R. S. Palais and S. Smale A generalized Morse theory. Bull. Amer. Math. Soc. 70 (1964), 165172.

(20)R. S. Rivlin Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformations. Phil. Trans. Roy. Soc. London A 240 (1948), 491508.

(22)R. T. Rockafellar Convex analysis (Princeton University Press, 1970).

(23)K. Sawyers and R. S. Rivlin Bifurcation conditions for a thick elastic plate under thrust. Int. J. Solids and Structures 10 (1974), 483501.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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