Skip to main content
Log in

Stability and related results in linear elastodynamics on unbounded regions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  1. S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand, 1965.

  2. L. Brun, Méthodes énergétiques dans les systèmes évolutifs linéaires. Deuxième partie: Théorèmes d'unicité. J. de Mech. 8, 167–192, 1969.

    Google Scholar 

  3. B. Carbonaro, & R. Russo, The uniqueness problem in the dynamical theory of linear hyperelasticity in unbounded domains. Atti VII Congresso Dell'AIMETA Vol. II, 47–55, 1984.

    Google Scholar 

  4. B. Carbonaro & R. Russo, A uniqueness theorem for a linear hyperbolic partial differential system of the second order. Comm. Partial Diff. Eqs. 9, 49–62, 1984.

    Google Scholar 

  5. B. Carbonaro & R. Russo, Energy inequalities and the domain of influence theorem in classical elastodynamics. J. Elast. 14, 163–174, 1984.

    Google Scholar 

  6. B. Carbonaro & R. Russo, On Graffi's reciprocal theorem in unbounded domains. J. Elast. 15, 35–42, 1985.

    Google Scholar 

  7. B. Carbonaro & R. Russo, On the work and energy theorem for unbounded elastic bodies. J. Elast. 15, 125–132, 1985.

    Google Scholar 

  8. B. Carbonaro & R. Russo, Some results on incompressible linear hyperelastic media. Quart. J. Mech. Appl. Math. 38, 507–520, 1985.

    Google Scholar 

  9. B. Carbonaro & R. Russo, Sharp uniqueness classes in linear elastostatics, submitted for publication.

  10. P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations. J. Math. Anal. Appl. 12, 401–414, 1973.

    Google Scholar 

  11. S. Chirita, Hölder continuous dependence in nonlinear elasticity. Proc. Roy. Soc. Edin. A 102, 49–60, 1986.

    Google Scholar 

  12. G. Duvaut & J.-L. Lions, Les Inéquations en Mécanique et en Physique. Dunod, 1972.

  13. G. Fichera, Sull'existenza e sul calcolo della soluzioni dei problemi al contorno relativi all'equilibrio di un corpo elastico. Ann. Scuolo Norm. Sup. Pisa 4, 35–99, 1950.

    Google Scholar 

  14. R. Finn, On the steady-state solutions of the Navier-Stokes equation. Acta Math. 105, 197–244, 1961.

    Google Scholar 

  15. G. P. Galdi & S. Rionero, Continuous dependence theorems in linear elasticity on exterior domains. Int. J. Eng. Sci. 17, 521–526, 1979.

    Google Scholar 

  16. G. P. Galdi & S. Rionero, On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions. J. Elast. 10, 333–340, 1980.

    Google Scholar 

  17. G. P. Galdi & S. Rionero, Continuous data dependence in linear elastodynamics on bounded domains without definiteness conditions on the elasticities. Proc. Roy. Soc. Edin. A 93, 299–306, 1983.

    Google Scholar 

  18. G. P, Galdi & S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity. Lecture Notes in Mathematics, No. 1134. Springer-Verlag, 1985.

  19. G. P. Galdi, R. J. Knops & S. Rionero, Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems. Math. Proc. Camb. Phil. Soc. 99, 357–366, 1986.

    Google Scholar 

  20. M. E. Gurtin & E. Sternberg, Theorems in linear elastostatics for exterior domains. Arch. Rational Mech. Anal. 8, 99–119, 1961.

    Google Scholar 

  21. K. B. Howell, Uniqueness in linear elasticity for problems involving unbounded regions. J. Elast. 10, 407–428, 1980.

    Google Scholar 

  22. D. S. Jones, An exterior problem in elastodynamics. Math. Proc. Camb. Phil. Soc. 96, 173–182, 1984.

    Google Scholar 

  23. D. S. Jones, A uniqueness theorem in elastodynamics. Quart. J. Math. Appl. Math. 37, 121–142, 1984.

    Google Scholar 

  24. R. J. Knops & L. E. Payne, Uniqueness Theorems in Linear Elasticity. Springer Tracts in Natural Philosophy No. 19. Springer-Verlag, 1971.

  25. R. J. Knops & L. E. Payne, Some uniqueness and continuous dependence theorems for nonlinear elastodynamics in exterior domains. Applic. Anal. 15, 33–51, 1983.

    Google Scholar 

  26. R. J. Knops, L. E. Payne & S. Rionero, Saint-Venant's Principle on unbounded domains. Proc. Roy. Soc. Edin. A 115, 319–336, 1990.

    Google Scholar 

  27. R. J. Knops & E. W. Wilkes, Theory of elastic stability. Handbuch der Physik VIa/3, 125–302, Springer, Berlin 1973.

    Google Scholar 

  28. V. A. Kondratiev & O. A. Oleinik, On the asymptotics at infinity of solutions of elliptic systems with constant coefficients. Uspekhi Mat. Nauk. 40, 233, 1985.

    Google Scholar 

  29. V. A. Kondratiev & O. A. Oleinik, On the behaviour at infinity of solutions of elliptic systems with finite energy integral. Arch. Rational Mech. Anal. 99, 75–89, 1987.

    Google Scholar 

  30. V. A. Kondratiev & O. A. Oleinik, Asymptotic properties of solutions of the elasticity system. Applications of Multiple Scaling in Mechanics (edited by P. G. Ciarlet and E. Sanchez-Palencia). Masson, 1987, 188–205.

  31. V. A. Kondratiev & O. A. Oleinik, Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities. Uspekhi Mat. Nauk 43, 55–98, 1988. Engl. Transl. in Russ. Math. Surveys 43, 65–119, 1988.

    Google Scholar 

  32. V. A. Kondratiev & O. A. Oleinik, Korn's type inequality for a class of unbounded domains and applications to boundary-value problems in elasticity. Elasticity. Mathematical Methods and Applications, The Ian N Sneddon 70th Birthday Volume. 211–233, Edited by G. Eason and R. W. Ogden, Ellis Horwood, 1990.

  33. V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili & T. V. Burchuladze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. Ed. by V. D. Kupradze. North-Holland, 1979.

  34. J.-L. Lions & F. Magenes, Non-Homogeneous Boundary-Value Problems and Applications. Vol. 1. Springer-Verlag 1972.

  35. C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation. Comm. Pure Appl. Math. 14, 561–568, 1961.

    Google Scholar 

  36. O. A. Oleinik & G. A. Yosifian, On the asymptotic behaviour at infinity of solutions in linear elasticity. Arch. Rational Mech. Anal. 78, 29–53, 1982.

    Google Scholar 

  37. R. Russo, On uniqueness for the displacement problem in the dynamical theory of elasticity. J. Elast. 15, 117–124, 1985.

    Google Scholar 

  38. R. Russo, On the stability of solutions to the system of classical elastodynamics. Z. angew. Math. Phys. 37, 124–133, 1986.

    Google Scholar 

  39. R. Russo, Continuous data dependence and uniqueness in homogeneous linear elastostatics. Boll. Un. Mat. Ital. (6) 5-A, 227–234, 1986.

    Google Scholar 

  40. R. Russo, An extension of the basic theorems of linear elastostatics to exterior domains. Ann. Univ. Ferrara 34, 101–119, 1988.

    Google Scholar 

  41. L. T. Wheeler, Some results in the linear dynamical theory of anisotropic elastic solids. Quart. Appl. Math. 38, 91–101, 1970.

    Google Scholar 

  42. L. T. Wheeler, On the uniqueness of solutions to the displacement problem in linear elastodynamics. J. Elast. 1, 121–124, 1971.

    Google Scholar 

  43. L. T. Wheeler & E. Sternberg, Some theorems in classical elastodynamics. Arch. Rational Mech. Anal. 31, 51–90, 1968.

    Google Scholar 

  44. C. A. Wilcox, Initial boundary value problems for linear hyperbolic partial differential equations of second-order. Arch. Rational Mech. Anal. 10, 361–400, 1962.

    Google Scholar 

  45. C. A. Wilcox, The domain of dependence inequality for symmetric hyperbolic systems, Bull. Amer. Math. Soc. 70, 149–154, 1964.

    Google Scholar 

  46. C. A. Wilcox, Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics. J. Elast. 9, 221–243, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by K. R. Rajagopal

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knops, R.J., Rionero, S. & Russo, R. Stability and related results in linear elastodynamics on unbounded regions. Arch. Rational Mech. Anal. 115, 179–200 (1991). https://doi.org/10.1007/BF00375226

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00375226

Keywords

Navigation