Mathematical Handbook Korn And Korn Pdf

Abstract

The authors discuss the existence and uniqueness up to isometries of E ⁿ of immersions ϕ: Ω ⊂ R ⁿ → E ⁿ with prescribed metric tensor field (g _ij ): Ω → S ⁿ _> , and discuss the continuity of the mapping (g _ij ) → ϕ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping (g _ij ) → ϕ can be obtained by means of nonlinear Korn inequalities is shown.

References

1. [1]

   Abraham, R., Marsden, J. E. and Ratiu, T., Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988.

   Book  MATH  Google Scholar

2. [2]

   Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.

   MATH  Google Scholar

3. [3]

   Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.

   MATH  Google Scholar

4. [4]

   Choquet-Bruhat, Y., Dewitt-Morette, C. and Dillard-Bleick, M., Analysis, Manifolds and Physics, North-Holland, Amsterdam, 1977.

   MATH  Google Scholar

5. [5]

   Ciarlet, P. G., Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013.

   MATH  Google Scholar

6. [6]

   Ciarlet, P. G. and Larsonneur, F., On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl., 81, 2002, 167–185.

   MathSciNet  Article  MATH  Google Scholar

7. [7]

   Ciarlet, P. G. and Laurent, F., Continuity of a deformation as a function of its Cauchy-Green tensor, Arch. Rational Mech. Anal., 167, 2003, 255–269.

   MathSciNet  Article  MATH  Google Scholar

8. [8]

   Ciarlet, P. G. and Mardare, C., Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., 83, 2004, 811–843.

   MathSciNet  Article  MATH  Google Scholar

9. [9]

   Ciarlet, P. G. and Mardare, C., Nonlinear Korn inequalities, J. Math. Pures Appl., 104, 2015, 1119–1134.

   MathSciNet  Article  MATH  Google Scholar

10. [10]

    Ciarlet, P. G. and Mardare, C., Inégalités de Korn non linéaires dans Rn, avec ou sans conditions aux limites, C. R. Acad. Sci. Paris, Ser. I, 353, 2015, 563–568.

    MathSciNet  Article  Google Scholar

11. [11]

    Ciarlet, P. G. and Mardare, S., Nonlinear Korn inequalities in Rn and immersions in W2,p,p > n,considered as functions of their metric tensors in W1,p, J. Math. Pures Appl., 105, 2016, 873–906.

    MathSciNet  Article  MATH  Google Scholar

12. [12]

    Ciarlet, P. G. and Mardare, S., Une inégalité de Korn non linéaire dans W2,p, p > n, C. R. Acad. Sci. Paris, Sér. I, 353, 2015, 905–911.

    MathSciNet  Article  MATH  Google Scholar

13. [13]

    Conti, S., Low-Energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns, Habilitationsschrift, Universität Leipzig, 2004.

    MATH  Google Scholar

14. [14]

    Friesecke, G., James, R. D. and Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., 55, 2002, 1461–1506.

    MathSciNet  Article  MATH  Google Scholar

15. [15]

    Malliavin, P., Géométrie Différentielle Intrinsèque, Hermann, Paris, 1992.

    MATH  Google Scholar

16. [16]

    Mardare, C., On the recovery of a manifold with prescribed metric tensor, Analysis and Applications, 1, 2003, 433–453.

    MathSciNet  Article  MATH  Google Scholar

17. [17]

    Mardare, S., On systems of first order linear partial differential equations with Lp coefficients, Adv. Diff. Eq., 12, 2007, 301–360.

    MathSciNet  MATH  Google Scholar

18. [18]

    Mardare, S., On isometric immersions of a Riemannian space with little regularity, Analysis and Applications, 2, 2004, 193–226.

    MathSciNet  Article  MATH  Google Scholar

19. [19]

    Necas, J., Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967.

    Google Scholar

20. [20]

    Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36, 1934, 63–89.

    MathSciNet  Article  MATH  Google Scholar

Download references

Author information

Affiliations

1. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

   Philippe G. Ciarlet

2. Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, CNRS-UMR 7598, F-75005, Paris, France

   Cristinel Mardare

3. Laboratoire de Mathématiques Raphaël Salem UMR 6085 CNRS-Université de Rouen Avenue de l'Université, BP.12 Technopôle du Madrillet, F76801, Saint-Etienne-du-Rouvray, France

   Sorin Mardare

Corresponding author

Correspondence to Philippe G. Ciarlet.

Additional information

Dedicated to Haïm Brezis on the occasion of his 70th birthday

This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, China (Nos. 9041637, CiyuU100711).

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Ciarlet, P.G., Mardare, C. & Mardare, S. Recovery of immersions from their metric tensors and nonlinear Korn inequalities: A brief survey. Chin. Ann. Math. Ser. B 38, 253–280 (2017). https://doi.org/10.1007/s11401-016-1070-5

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• Received: 01 June 2015

• Revised: 07 October 2015

• Published: 05 January 2017

• Issue Date: January 2017

• DOI : https://doi.org/10.1007/s11401-016-1070-5

Keywords

• Isometric immersions
• Nonlinear Korn inequalities
• Metric tensor

2000 MR Subject Classification

• 74B20
• 53C24

Mathematical Handbook Korn And Korn Pdf

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