Skip to main content

An optimization problem for the Biharmonic equation with Sobolev conditions

We state and solve an optimization problem about distribution of several supporting points under a Kirchhoff plate clamped along the boundary: the biharmonic equation is supplied with the Dirichlet boundary conditions and point Sobolev conditions. Some open questions are formulated. Bibliography: 23 titles.

References

  1. S. G. Mikhlin, Variational Methods in Mathematical Physics [in Russian], Nauka, Moscow (1970); English transl.: Pergamon Press, Oxford (1964).

    Google Scholar 

  2. S. Germain, Recherches sur la Théorie des Surfaces Élastiques, Courcier, Paris (1821).

    Google Scholar 

  3. G. R. Kirchhoff, “Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe,” J. Reine Angew. Math. 40, 51–88 (1850); 56, 285–313 (1859).

  4. D. Morgenstern, “Herleitung der Plattentheorie aus der dreidimensionallen Elastozit ätstheorie,” Arch. Ration. Mech. Anal. 4, 145–152 (1959).

    MathSciNet  MATH  Article  Google Scholar 

  5. B. A. Shoikhet, “On asymptotically exact equations of thin plates of complex structure” [in Russian], Prikl. Mat. Mekh. 37, 914–924 (1973); English transl.: J. Appl. Math. Mech. 37, 867–877 (1974).

    MathSciNet  Article  Google Scholar 

  6. P. G. Ciarlet and P. Destuynder, “A justification of the two-dimensional plate model,” J. Mec. Paris 18, 315–344 (1979).

    MathSciNet  MATH  Google Scholar 

  7. S. A. Nazarov, “Asymptotic analysis of an arbitrary anisotropic plate of variable thickness (a shallow shell)” [in Russian], Mat. Sb. 191, No 7, 129–159 (2000); English transl.: Sb. Math. 191, No. 7–8, 1075–1106 (2000).

    MathSciNet  MATH  Article  Google Scholar 

  8. D. Percivale and P. Podio-Guidugli, “A general linear theory of elastic plates and its variational validation,” Boll. Unione Mat. Ital. (9) 2, No. 2, 321–341 (2009).

    MathSciNet  MATH  Google Scholar 

  9. P. G. Ciarlet, Mathematical Elasticity, II: Theory of Plates, North-Holland, Amsterdam (1997).

    Google Scholar 

  10. P. Destuynder, Une Théorie Asymptotique des Plaques Minces en Elasticité Linéaire, Masson, Paris (1986).

    MATH  Google Scholar 

  11. S. A. Nazarov, Asymptotic Theory of Thin Plates and Rods. 1. Dimension Reduction and Integral Estimates [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (2002).

    Google Scholar 

  12. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973); English transl.: Springer, New York etc. (1985).

    Google Scholar 

  13. J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. I, Springer, Berlin etc. (1972).

    Google Scholar 

  14. V. A. Kondrat’ev, “Boundary value problems for elliptic equations in domains with conical and angular points” [in Russian], Tr. Mosk. Mat. O-va 16, 219–292 (1963); English transl.: Trans. Moscow Math. Soc. 16, 227–313 (1967).

    MATH  Google Scholar 

  15. S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundaries [in Russian], Nauka, Moscow (1991); English transl.: Walter de Gruyter, Berlin etc. (1994).

    Book  Google Scholar 

  16. V. A. Kozlov, V. G, Maz’ya, J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities Am. Math. Soc., Providence, RI (1997).

    MATH  Google Scholar 

  17. W. G. Maz’ya, S. A., Nazarov, B. A. Plamenevskii, Asymptotische Theorie Elliptischer Randwertaufgaben in Singulär Gestörten Gebieten. I [in German], Akademie-Verlag, Berlin (1991); English transl.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. I, Birkhäuser, Basel (2000).

    Google Scholar 

  18. G. Buttazzo and S. A. Nazarov, “Optimal location of support points in the Kirchhoff plate,” In: Proceedings of the Meeting “Variational Analysis and Aerospace Engineering II”. Erice 2010, Springer. [To appear]

  19. G. Buttazzo, E. Oudet, and E. Stepanov, “Optimal transportation problems with free Dirichlet regions,” In: Variational Methods for Discontinuous Structures. Cernobbio 2001, pp. 41–65, Birkhäuser, Basel (2002).

    Google Scholar 

  20. G. Bouchitté, C. Jimenez, and R. Mahadevan, “Asymptotic analysis of a class of optimal location problems,” J. Math. Pures Appl. (9) 95 (4) 382–419 (2011).

    MathSciNet  MATH  Google Scholar 

  21. G. Buttazzo, F. Santambrogio, and N. Varchon, “Asymptotics of an optimal compliancelocation problem,” ESAIM Control Optim. Calc. Var. 12, 752–769 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  22. V. G. Maz’ya and B. A. Plamenevskii, “Estimates in L p and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary” Math. Nachr. 77, 25–82 (1977).

    Google Scholar 

  23. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems [in Russian], Nauka, Moscow (1989); English transl.: Am. Math. Soc., Providence, RI (1992).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. A. Nazarov.

Additional information

Dedicated to Vasilii Vasil’evich Zhikov

Translated from Problems in Mathematical Analysis 58, June 2011, pp. 69–77.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Buttazzo, G., Nazarov, S.A. An optimization problem for the Biharmonic equation with Sobolev conditions. J Math Sci 176, 786 (2011). https://doi.org/10.1007/s10958-011-0436-1

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s10958-011-0436-1

Keywords

  • Optimal Location
  • Green Function
  • Dirichlet Problem
  • Elliptic Boundary
  • Biharmonic Equation