A Necessary and Sufficient Condition for the Existence of Absolute Minimizers for Energy Functionals with Scale Invariance
In (Haidar 2009) and (Haidar 2007), we constructed models in higher dimensional nonlinear hyper-elasticity that establish that strong ellipticity of the stored energy does not imply that all solutions to the corresponding equilibrium equations which pass through the origin and have finite energy are trivial. While that work, by itself, settled a longstanding problem posed by J. Ball in 1982 (Ball 1982), it also shed new light on another central, very difficult problem of nonlinear elasticity, namely, that of regularity of weak equilibria and material instabilities.
KeywordsLagrange Equation Nonlinear Partial Differential Equation Absolute Minimizer Nonlinear Elasticity Energy Functional
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- [AF00]Ambriosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variations and Free Discontinuity Problems, Clarendron Press, Oxford University Press, New York (2000). Google Scholar
- [Er83]Ericksen, J.L.: Ill-posed problems in thermoelasticity theory, in: Systems of Nonlinear Partial Differential Equations (Editor: J.M. Ball), Reidel, Dordrecht (1983). Google Scholar
- [GeFo63]Gelfand, I.M., Fomin, S.V.: Calculus of Variations, Prentice-Hall (1963). Google Scholar
- [Ha00]Haidar, S.M.: Existence and regularity of weak solutions to the displacement boundary value problem of nonlinear elastostatics, in: Integral Methods in Science and Engineering, CRC Press, 161–166 (2000). Google Scholar
- [Ha07]Haidar, S.M.: Convexity conditions in uniqueness and regularity of equilibria in nonlinear elasticity, in: Integral Methods in Science and Engineering: Techniques and Applications (Editors: C. Constanda, S. Potapenko), Springer–Birkhäuser, Boston, MA, 109–119 (2007). Google Scholar
- [Ha09]Haidar, S.M.: On J. Ball’s fundamental existence theory and regularity of weak equilibria in nonlinear radial hyperelasticity, in: Integral Methods in Science and Engineering: Techniques and Applications, vol. 1 (Editors: C. Constanda, M. Pérez), Springer–Birkhäuser, Boston, MA, 161–171 (2009). Google Scholar
- [Lo77]Logan, J.D.: Invariant Variational Principles, Academic Press (1977). Google Scholar