Skip to main content
SpringerLink
Log in

A mixed problem of linear elastodynamics

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

This paper is devoted to a semigroup approach to an initial-boundary value problem of linear elastodynamics in the case where the boundary condition is a regularization of the genuine mixed displacement-traction boundary condition. More precisely, it is a smooth linear combination of displacement and traction boundary conditions, but is not equal to the pure traction boundary condition. Some previous results with mixed displacement-traction boundary condition are due to Inoue and Ito. The crucial point in our semigroup approach is to generalize the classical variational approach to the degenerate case, by using the theory of fractional powers of analytic semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. A. Adams and J. J. F. Fournier, Sobolev spaces, second edition. Academic Press, Amsterdam, Heidelberg, New York, Oxford (2003)

  2. Bergh J., Löfström J.: Interpolation spaces, An introduction. Springer, Berlin, New York (1976)

    Book  MATH  Google Scholar 

  3. Chazarain J., Piriou A.: Introduction à la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)

    MATH  Google Scholar 

  4. P. G. Ciarlet, Mathematical elasticity, vol. I. Stud. Math. Appl. no. 20. North-Holland, Amsterdam, New York, Oxford, Tokyo (1988)

  5. G. Duvaut et J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972)

  6. Fujiwara D.: Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad. Ser. A Math. Sci. 43, 82–86 (1967)

    Article  MATH  Google Scholar 

  7. Goldstein J.A.: Semigroups and second-order differential equations. J. Func. Anal. 4, 50–70 (1969)

    Article  MATH  Google Scholar 

  8. Goldstein J.A.: Semigroups of linear operators and applications. Oxford University Press, New York (1985)

    MATH  Google Scholar 

  9. K. Hayashida, On a mixed problem for hyperbolic equations with discontinuous boundary conditions. Publ. RIMS, Kyoto Univ. 7, 57–67 (1971/72)

    Google Scholar 

  10. E. Hille and R. S. Phillips, Functional analysis and semi-groups. Am. Math. Soc. Colloq. Publ. vol. 31. Amer. Math. Soc., Providence, Rhode Island (1957)

  11. L. Hörmander, The analysis of linear partial differential operators III. Grundlehren Math. Wiss. Springer, Berlin Heidelberg, New York, Tokyo (1985)

  12. Ibuki K.: On the regularity of solutions of a mixed problem for hyperbolic equations of second order in a domain with corners. J. Math. Kyoto Univ. 16, 167–183 (1976)

    MathSciNet  MATH  Google Scholar 

  13. Ikawa M.: Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20, 580–608 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  14. Inoue A.: On a mixed problem for □ with discontinuous boundary condition (I). J. Fac. Sci. Univ. Tokyo Sect. IA 21, 85–92 (1974)

    MATH  Google Scholar 

  15. Inoue A.: On a mixed problem for d’Alembertian with a mixed boundary condition - an example of moving boundary. Publ. RIMS, Kyoto Univ. 11, 339–401 (1976)

    Article  MATH  Google Scholar 

  16. Ito H.: On a mixed problem of linear elastodynamics with time-dependent discontinuous boundary condition. Osaka J. Math. 27, 667–707 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Marsden J.E., Hughes T.J.R.: Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs, New Jersey (1983)

    MATH  Google Scholar 

  18. Stein E.M.: Singular integrals and differentiability properties of functions. Princeton Math. Ser. Princeton University Press, Princeton, New Jersey (1970)

    MATH  Google Scholar 

  19. K. Taira, Analytic semigroups and semilinear initial-boundary value problems. London Math. Soc. Lecture Note Ser. vol. 223. Cambridge University Press, Cambridge (1995)

  20. Taira K.: Introduction to boundary value problems of nonlinear elastostatics. Tsukuba J. Math. 32, 67–138 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Taira K.: Degenerate elliptic boundary value problems with asymmetric nonlinearity. J. Math. Soc. Japan 62, 431–465 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Triebel H.: Theory of function spaces. Monogr. Math. Birkhäuser Verlag, Basel (1983)

    Book  Google Scholar 

  23. T. Valent, Boundary value problems of finite elasticity. Springer Tracts Nat. Phil. no 31. Springer, New York, Berlin, Heidelberg, Tokyo (1988)

  24. Weiss B.: Abstract vibrating systems. J. Math. Mech. 17, 241–255 (1967)

    MathSciNet  MATH  Google Scholar 

  25. K. Yosida, Functional analysis, sixth edition. Grundlehren Math. Wiss. Springer, Berlin, Heidelberg, New York (1980)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, 305–8571, Japan

    Kazuaki Taira

Authors
  1. Kazuaki Taira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuaki Taira.

Additional information

Dedicated to Professor Mitsuru Ikawa on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taira, K. A mixed problem of linear elastodynamics. J. Evol. Equ. 13, 481–507 (2013). https://doi.org/10.1007/s00028-013-0187-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-013-0187-1

Mathematics Subject Classification (2010)

Keywords

Search

Navigation