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A history of the Cosserat spectrum

  • Conference paper
The Maz’ya Anniversary Collection

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 109))

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Abstract

The paper is a brief survey of the 100-year history of the Cosserat spectrum in elastostatics which was first studied by Eugène and François Cosserat between 1898 and 1901 and later by Vladimir Maz’ya and Solomon Mikhlin.

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References

  1. Eugène et François Cosserat, Sur les équations de la théorie de l’élasticité, C.R. Acad. Sci. (Paris) 126 (1898), 1089–1091.

    Google Scholar 

  2. Eugène et François Cosserat, Sur les fonctions potentielles de 1a théorie de l’élasticité, C.R. Acad. Sci. (Paris) 126 (1898), 1129–1132.

    Google Scholar 

  3. Eugène et François Cosserat, Sur la déformation infiniment petite d’un ellipsoide élastique, C.R. Acad. Sci. (Paris) 127 (1898), 315–318.

    Google Scholar 

  4. Eugène et François Cosserat, Sur la solution des équations de l’élasticité dans le cas où les valeurs des inconnues à la frontière sont données, C.R. Acad. Sci. (Paris) 133 (1901), 145–147.

    Google Scholar 

  5. Eugène et François Cosserat, Sur une application des fonctions potentielles de 1a théorie de l’élasticité, C.R. Acad. Sci. (Paris) 133 (1901), 210–213.

    Google Scholar 

  6. Eugène et François Cosserat, Sur la déformation infiniment petite d’un corps élastique soumis à des forces données, C.R. Acad. Sci. (Paris) 133 (1901), 271–273.

    Google Scholar 

  7. Eugène et François Cosserat, Sur la déformation infiniment petite d’une enveloppe sphérique élastique, C.R. Acad. Sci. (Paris) 133 (1901), 326–329.

    Google Scholar 

  8. Eugène et François Cosserat, Sur la déformation infiniment petite d’un ellipsoide élastique soumis à des efforts données sur la frontière, C.R. Acad. Sci. (Paris) 133 (1901), 361–364.

    Google Scholar 

  9. Eugène et François Cosserat, Sur un point critique particulier de la solution des équations de l’élasticité dans le cas où les efforts sur la frontière sont données, C.R. Acad. Sci. (Paris) 133 (1901), 382–384.

    Google Scholar 

  10. Dictionary of Scientific Biography, Charles Scribner’s Sons, New York, 1971.

    Google Scholar 

  11. G. Grubb, G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann. 227 (1977), 247–276.

    Article  MathSciNet  Google Scholar 

  12. G. Geymonat, E. Sanchez-Palencia, On the vanishing viscosity limit for acoustic phenomena in a bounded region, Archiv for Rat. Mech. and Anal., 75 (1981), 257–268.

    Google Scholar 

  13. M. E. Gurtin, The linear theory of elasticity. In: Flügge S., Truesdell C. (ed.) Handbuch der Physik, vol. YIa/2, pp. 347–424, Springer, Berlin, 1972.

    Google Scholar 

  14. R. J. Knops, L. E. Payne, Uniqueness theorems in linear elasticity, Springer, New York, 1971.

    MATH  Google Scholar 

  15. A. N. Kozhevnikov, On the second and third boundary value problems of the static elasticity theory, Sov. Math. Dokl., 38 (1989), 427–430.

    Google Scholar 

  16. A. N. Kozhevnikov, The basic boundary value problems of the static elasticity theory and their Cosserat spectrum, Mathematische Zeitschrift, 213 (1993), 241–274.

    Article  MathSciNet  Google Scholar 

  17. A. Kozhevnikov, On the first stationary boundary-value problem of elasticity in weighted Sobolev spaces in exterior domains of ℝ3, Applied Mathematics and Optimization, 34 (1996), 183–190.

    Article  MathSciNet  Google Scholar 

  18. A. Kozhevnikov, On a lower bound for the Cosserat eigenvalues, Applicable Analysis (to appear).

    Google Scholar 

  19. A. Kozhevnikov, T. Skubachevskaya, Some applications of pseudo-differential operators to elasticity, Hokkaido Math. J., 26, no. 2 (1997), 297–322.

    MathSciNet  MATH  Google Scholar 

  20. A. Kozhevnikov, O. Lepsky, Power Series Solutions to Basic Stationary Boundary Value Problems of Elasticity, Integral Equations and Operator Theory, 31, (1998), 449–469.

    Article  MathSciNet  Google Scholar 

  21. V. D. Kupradze, T. G. Gegelia, T. V. Burchuladze, H. O. Basheleishvili, Three-dimensional problems of elasticity and thermoelasticity, North-Holland, Amsterdam, New York, 1979.

    MATH  Google Scholar 

  22. Levitin, M.R., On the spectrum of a generalized Cosserat problem, C. R. Acad. Sci. Paris, 315, Série I (1992), 925–930.

    MathSciNet  MATH  Google Scholar 

  23. L. Lichtenstein, Über die erste Randwertaufgabe der Elastizitätstheorie, Math. Zeit. 20 (1924), 21–28.

    Article  MATH  Google Scholar 

  24. A. E. H. Love, A treatise on the mathematical theory of elasticity, Cambridge University Press, London, 1934.

    Google Scholar 

  25. V. G. Mazya, S. G. Mikhlin, On the Cosserat spectrum of the equations of the theory of elasticity, Vestnik Leningrad Univ. Math. No. 3 (1967), 58–63.

    Google Scholar 

  26. S. G. Mikhlin, On Cosserat functions. In: Problemy matem. analiza. Kraev. zadachi i integr. uravneniya (Problems of mathematical analysis. Boundary value problems and integral equations) Izdat. Leningrad Univ. 1966, 56-59.

    Google Scholar 

  27. S. G. Mikhlin, Further investigation of Cosserat functions, Vestnik Leningrad. Univ., no. 2 (1967), 98–102.

    Google Scholar 

  28. S. G. Mikhlin, Some properties of the Cosserat spectrum of spatial and plane problems of the theory of elasticity, Vestnik Leningrad. Univ., no. 7 (1970), 31–45.

    Google Scholar 

  29. S. G. Mikhlin, The Cosserat spectrum of static problems of the theory of elasticity and its application. In: Problemy mekhaniki tverdogo deformiruemogo tela (Problems of the mechanics of a solid deformable body), Sudostroenie, Leningrad 1970, 265-271.

    Google Scholar 

  30. S. G. Mikhlin, The Cosserat spectrum of problems of the theory of elasticity for infinite domains. In: Issledovaniya po uprugosti i plastichnosti (Investigations on elasticity and plasticity), 1973, no. 9.

    Google Scholar 

  31. S. G. Mikhlin, The spectrum of a family of operators in the theory of elasticity, Russian Math. Surveys., 28, no. 3 (1973), 45–88.

    Article  MATH  Google Scholar 

  32. Mikhlin S.G., Morozov, N.F., Paukshto, M.V., The Integral Equations of the Theory of Elasticity, Teubner, Stuttgart, 1995.

    MATH  Google Scholar 

  33. X. Markenscoff, M. Paukshto, On the Cavities and Rigid Inclusions Correspondence and the Cosserat Spectrum, Math. Nachr. 177 (1996), 183–188.

    Article  MathSciNet  MATH  Google Scholar 

  34. X. Markenscoff, M. Paukshto, The Cosserat spectrum in the theory of elasticity and applications, Proc. R. Soc. Lond. A (1998) 454, 631–643.

    Article  Google Scholar 

  35. M. Paukshto, On Some Applications of Integral Equations in Elasticity, Proc.of 4th Int. Conference in Integral Methods in Science and Engineering, IMSE 96, Oulu, Finland.

    Google Scholar 

  36. M. C. Pelissier, Résolution numérique de quelques problèmes raides en mécanique des milieux faiblement compressibles, Calcolo, 12 (1975), 275–314.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam, New York, Oxford (1979).

    Google Scholar 

  38. E. Trefftz, Die mathematische Theorie der Elastizität, Teubner, Leipzig 1931.

    Google Scholar 

  39. W. Veite, On inequalities of Friedrichs and Babuška-Aziz in Dimension Three, Zeitschrift für Analysis und ihre Anwendungen, 17 (1998), 843–857.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    Alexander Kozhevnikov

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  1. Alexander Kozhevnikov
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Rostock, D-18051, Rostock, Germany

    Jürgen Rossmann , Peter Takáč  & Günther Wildenhain ,  & 

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Kozhevnikov, A. (1999). A history of the Cosserat spectrum. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_16

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  • DOI: https://doi.org/10.1007/978-3-0348-8675-8_16

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9726-6

  • Online ISBN: 978-3-0348-8675-8

  • eBook Packages: Springer Book Archive

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