New nonlinear estimates for surfaces in terms of their fundamental forms. (Nouvelles estimations pour des surfaces en fonction de leurs formes fondamentales.)(English. French summary)Zbl 1361.53004

Consider surfaces $$\theta$$, $$\tilde{\theta}$$ which are defined over the same domain $$\omega$$ and immersed into $$\mathbb{R}^3$$. Their normal vector fields are denoted by $$a_3$$ and $$\tilde{a}_3$$, respectively. The authors bound the sum $\| \tilde{\theta} - \theta \|_{W^{1,p}(\omega)} + \| a_3(\tilde{\theta}) - a_3(\theta) \|_{W^{1,p}(\omega)}$ and the infimum over all isometries $$r$$ of $\| r \circ \tilde{\theta} - \theta \|_{W^{1,p}(\omega)} + \| a_3(r \circ \tilde{\theta}) - a_3(\theta) \|_{W^{1,p}(\omega)}.$ The bounds are given in terms of Sobolev norms on differences of first, second, and/or third fundamental forms of these surfaces and constants that depend on $$\omega$$ and two parameters $$p > 1$$, $$q \geq 1$$ with $$p/2 \leq q \leq p$$.
Estimates of this type are particularly useful in connection with the Koiter shell model of nonlinear elasticity theory where they allow to control the magnitude of the surface deformation in terms of the strain energy.
This article only provides definitions, results, and sketches of proofs. More details and extension to higher Sobolev norms are announced for a forthcoming paper.

MSC:

 53A05 Surfaces in Euclidean and related spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 74K25 Shells 74B20 Nonlinear elasticity
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References:

 [1] Adams, R. A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] Bernadou, M.; Ciarlet, P. G., Sur l’ellipticité du modèle linéaire de coques de W.T. Koiter, (Glowinski, R.; Lions, J.-L., Computing Methods in Applied Sciences and Engineering, Lect. Notes Econ. Math. Syst., vol. 134, (1976), Springer-Verlag Heidelberg), 89-136 · Zbl 0356.73066 [3] Bernadou, M.; Ciarlet, P. G.; Miara, B., Existence theorems for two-dimensional linear shell theories, J. Elasticity, 34, 111-138, (1994) · Zbl 0808.73045 [4] Blouza, A.; Le Dret, H., Existence and uniqueness for the linear Koiter model for shells with little regularity, Q. Appl. Math., 57, 317-337, (1999) · Zbl 1025.74020 [5] Ciarlet, P. G., An introduction to differential geometry with applications to elasticity, (2005), Springer Dordrecht, The Netherlands · Zbl 1086.74001 [6] Ciarlet, P. G.; Gratie, L.; Mardare, C., A nonlinear korn inequality on a surface, J. Math. Pures Appl., 85, 2-16, (2006) · Zbl 1094.53001 [7] Ciarlet, P. G.; Mardare, C., On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., 83, 1-15, (2004) · Zbl 1050.74030 [8] Ciarlet, P. G.; Mardare, C., Nonlinear korn inequalities, J. Math. Pures Appl., 104, 1119-1134, (2015) · Zbl 1337.35144 [9] P.G. Ciarlet, M. Malin, C. Mardare, New estimates of the distance between two surfaces in terms of the distance between their fundamental forms, in preparation. · Zbl 1361.53004 [10] Klingenberg, W., A course in differential geometry, (1978), Springer Berlin [11] Koiter, W. T., On the nonlinear theory of thin elastic shells, Proc. K. Ned. Akad. Wet., B69, 1-54, (1966) [12] Nečas, J., LES méthodes directes en théorie des équations elliptiques, (1967), Masson Paris · Zbl 1225.35003
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