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Journal of Convex Analysis 17 (2010), No. 1, 131--158
Copyright Heldermann Verlag 2010

Dual Variational Formulations for a Non-Linear Model of Plates

Fabio Botelho
Mathematics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.
Permanent address: Mathematics Department, Federal University of Pelotas, Pelotas-RS, Brasil

This article develops dual variational formulations for the two dimensional equations of the nonlinear elastic Kirchhoff-Love plate model. The first duality principle presented is the classical one and may be found in similar format in articles of J. J. Telega [C. R. Acad. Sci. Paris, Serie II-308 (1989) 1193-1198, 1313-1317] and D. Y. Gao [Quarterly of Applied Mathematics 48(2) (1990) 361-370]. It is worth noting that such results are valid only for positive definite membrane forces, however, we obtain new dual variational formulations which relax or even remove such constraints. Among our results we have a convex dual variational formulation which allows non positive definite membrane forces. In the last section, similarly to the Triality criterion introduced by D. Y. Gao ["Duality Principles in Nonconvex Systems. Theory, Methods and Applications", Kluwer, Dordrecht (2000)], we obtain sufficient conditions of optimality for the present case. Finally, the results are based on fundamental tools of Convex Analysis and also relevant for the developed theory is the concept of Legendre Transform, which can easily be analytically expressed for the mentioned model.

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