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

Published online by Cambridge University Press:  24 October 2008

J. M. Ball
J. M. Ball
Affiliation:
University of California, Berkeley
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 3 , May 1980 , pp. 501 - 513
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Antman, S. S. and Osborn, J. E.The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69 (1979), 231–262.Google Scholar
(2)Ball, J. M.Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
(3)Ball, J. M. Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, vol. 1, ed. Knops, R. J., (London, Pitman, 1977).Google Scholar
(4)Ball, J. M. 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, 13–25.CrossRefGoogle Scholar
(5)Ball, J. M. Remarques sur l'existence et la régularité des solutions d'élastostatique non-lineáire. (In the Press).Google Scholar
(6)Ball, J. M. In preparation.Google Scholar
(7)Ericksen, J. L.Equilibrium of bars. J. of Elasticity 5 (1975), 191–201.CrossRefGoogle Scholar
(8)Ericksen, J. L. Special topics in elastostatics. In Advances in Applied Mechanics 17 (1977) New York.Google Scholar
(9)Giusti, E. and Miranda, M.Un esempio di soluzioni discontinue per un problema diminimo relativo ad un integrale regolare del calcola delle variazioni. Boll. Unione Mat. Ital. Ser. 4. 1 (1968), 219–226.Google Scholar
(10)Hadamard, J.Leçons sur la propagation des ondes (Paris, Hermann, 1903).Google Scholar
(11)Hartman, P.Ordinary differential equations (New York, John Wiley, 1964).Google Scholar
(12)Hartman, P. and Olech, C.On global asymptotic stability of solutions of ordinary differential equations. Trans. Amer. Math. Soc. 104 (1962), 154–178.Google Scholar
(13)Knowles, J. K. and Sternberg, E.On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elasticity 5 (1975), 341–362.CrossRefGoogle Scholar
(14)Knowles, J. K. and Sternberg, E.On the failure of ellipticity of the equations for finite elastic plane strain. Arch. Rational Mech. Anal. 63 (1977), 321–326.CrossRefGoogle Scholar
(15)Knowles, J. K. and Sternberg, E.On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elasticity. J. Elasticity 9 (1978), 329–380.Google Scholar
(16)Necas, J. Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In Theory of non-linear operators (Berlin, Akademie-Verlag, 1977).Google Scholar
(17)Olech, C.On the global stability of autonomous systems in the plane. Contributions to Differential Equations 1 (1963), 389–400.Google Scholar
(18)Palais, R. S.Critical point theory and the minimax principle. Proc. Symp. Pure Math. 15, Amer. Math. Soc. Providence, R. I. (1970), 185–212.Google Scholar
(19)Palais, R. S. and Smale, S.A generalized Morse theory. Bull. Amer. Math. Soc. 70 (1964), 165–172.CrossRefGoogle Scholar
(20)Rivlin, R. S.Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformations. Phil. Trans. Roy. Soc. London A 240 (1948), 491–508.Google Scholar
(21)Rivlin, R. S.Stability of pure homogeneous deformations of an elastic cube under dead loading. Quart. Appl. Math. 32 (1974), 265–271.CrossRefGoogle Scholar
(22)Rockafellar, R. T.Convex analysis (Princeton University Press, 1970).Google Scholar
(23)Sawyers, K. and Rivlin, R. S.Bifurcation conditions for a thick elastic plate under thrust. Int. J. Solids and Structures 10 (1974), 483–501.CrossRefGoogle Scholar
(24)Truesdell, C. and Noll, W. The non-linear field theories of mechanics. In Handbuch der Physik, ed. Flugge, S., vol. 3, part 3, 1–590 (Berlin and New York, Springer-Verlag, 1965).Google Scholar