Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-25T06:13:10.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved estimates for continuous data dependence in linear elastodynamics

Published online by Cambridge University Press:  24 October 2008

R. J. Knops  and
L. E. Payne
R. J. Knops
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh
L. E. Payne
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York
Get access Rights & Permissions [Opens in a new window]

Extract

In a previous paper [6], the present authors established estimates for the continuous dependence of the solution on various data in the initial boundary value problem of linear elastodynamics on a bounded region of space. The main conclusion concerned continuous dependence on the body-force, but also it was shown how this result could be used to derive continuous dependence on the initial data, elasticities, boundary data and initial geometry. The method adopted was based upon logarithmic convexity arguments and hence led naturally to continuity in the sense of Hölder on compact sub-intervals of time. A special feature of the study entailed the lack of any sign-definiteness conditions on the elasticities which, of course, in the absence of any a priori constraint on the solution always gives rise to an ill-posed problem. (See, for instance, the comprehensive survey by Payne [10].)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 535 - 559
Copyright
Copyright © Cambridge Philosophical Society 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Brun, L.. Méthodes énergetiques dans les systemes évolutifs linéaires. Deuxième Partie: Théorèmes d'unicité. J. de Méchanique 8 (1969), 167–192.Google Scholar
[2]Galdi, G. P., Knops, R. J. and Rionero, S.. Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems. Math. Proc. Cambridge Philos. Soc. 99 (1986), 357–366.CrossRefGoogle Scholar
[3]Green, A. E.. A note on linear thermoelasticity. Mathematika 19 (1972), 69–75.CrossRefGoogle Scholar
[4]Knops, R. J. and Payne, L. E.. Stability in linear elasticity. Int. J. Sols. Structures 4 (1968), 1233–1242.CrossRefGoogle Scholar
[5]Knops, R. J. and Payne, L. E.. Uniqueness in classical elastodynamics. Arch. Rational Mech. Anal. 27 (1968), 349–355.CrossRefGoogle Scholar
[6]Knops, R. J. and Payne, L. E.. Continuous data dependence for the equations of classical elastodynamics. Proc. Cambridge Philos. Soc. 66 (1969), 481–491.CrossRefGoogle Scholar
[7]Knops, R. J. and Payne, L. E.. Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics. Arch. Rational Mech. Anal. 41 (1971), 363–398.CrossRefGoogle Scholar
[8]Levine, H. A.. An equipartition of energy theorem for weak solutions of evolutionary equations in Hilbert space: The Lagrange identity method. J. Diff. Eqns. 24 (1977), 197–210.CrossRefGoogle Scholar
[9]Naghdi, P. M. and Trapp, J. A.. A uniqueness theorem in the theory of Cosserat surface. J. Elasticity 2 (1972), 9–20.CrossRefGoogle Scholar
[10]Payne, L. E.. Improperly Posed Problems in Partial Differential Equations. Regional Conference Series in Applied Mathematics (SIAM, 1975).CrossRefGoogle Scholar
[11]Payne, L. E.. On stability and growth of solutions to second-order operator equations. In Mathematical Methods and Models in Mechanics (Banach Center Publication No. 15, Warsaw 1985), pp. 465–475.Google Scholar
[12]Payne, L. E.. On geometric and modeling perturbations in partial differential equations. In Proceedings of the L.M.S. Durham Symposium on Non-Classical Continuum Mechanics (Cambridge University Press, 1987), pp. 108–128.CrossRefGoogle Scholar
[13]Tartar, L.. Nonlinear Partial Differential Equations using Compactness Method. MRC Technical Summary Report No. 1584, University of Wisconsin (1976).Google Scholar