References
S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand, 1965.
L. Brun, Méthodes énergétiques dans les systèmes évolutifs linéaires. Deuxième partie: Théorèmes d'unicité. J. de Mech. 8, 167–192, 1969.
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.
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.
B. Carbonaro & R. Russo, Energy inequalities and the domain of influence theorem in classical elastodynamics. J. Elast. 14, 163–174, 1984.
B. Carbonaro & R. Russo, On Graffi's reciprocal theorem in unbounded domains. J. Elast. 15, 35–42, 1985.
B. Carbonaro & R. Russo, On the work and energy theorem for unbounded elastic bodies. J. Elast. 15, 125–132, 1985.
B. Carbonaro & R. Russo, Some results on incompressible linear hyperelastic media. Quart. J. Mech. Appl. Math. 38, 507–520, 1985.
B. Carbonaro & R. Russo, Sharp uniqueness classes in linear elastostatics, submitted for publication.
P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations. J. Math. Anal. Appl. 12, 401–414, 1973.
S. Chirita, Hölder continuous dependence in nonlinear elasticity. Proc. Roy. Soc. Edin. A 102, 49–60, 1986.
G. Duvaut & J.-L. Lions, Les Inéquations en Mécanique et en Physique. Dunod, 1972.
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.
R. Finn, On the steady-state solutions of the Navier-Stokes equation. Acta Math. 105, 197–244, 1961.
G. P. Galdi & S. Rionero, Continuous dependence theorems in linear elasticity on exterior domains. Int. J. Eng. Sci. 17, 521–526, 1979.
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.
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.
G. P, Galdi & S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity. Lecture Notes in Mathematics, No. 1134. Springer-Verlag, 1985.
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.
M. E. Gurtin & E. Sternberg, Theorems in linear elastostatics for exterior domains. Arch. Rational Mech. Anal. 8, 99–119, 1961.
K. B. Howell, Uniqueness in linear elasticity for problems involving unbounded regions. J. Elast. 10, 407–428, 1980.
D. S. Jones, An exterior problem in elastodynamics. Math. Proc. Camb. Phil. Soc. 96, 173–182, 1984.
D. S. Jones, A uniqueness theorem in elastodynamics. Quart. J. Math. Appl. Math. 37, 121–142, 1984.
R. J. Knops & L. E. Payne, Uniqueness Theorems in Linear Elasticity. Springer Tracts in Natural Philosophy No. 19. Springer-Verlag, 1971.
R. J. Knops & L. E. Payne, Some uniqueness and continuous dependence theorems for nonlinear elastodynamics in exterior domains. Applic. Anal. 15, 33–51, 1983.
R. J. Knops, L. E. Payne & S. Rionero, Saint-Venant's Principle on unbounded domains. Proc. Roy. Soc. Edin. A 115, 319–336, 1990.
R. J. Knops & E. W. Wilkes, Theory of elastic stability. Handbuch der Physik VIa/3, 125–302, Springer, Berlin 1973.
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.
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.
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.
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.
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.
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.
J.-L. Lions & F. Magenes, Non-Homogeneous Boundary-Value Problems and Applications. Vol. 1. Springer-Verlag 1972.
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.
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.
R. Russo, On uniqueness for the displacement problem in the dynamical theory of elasticity. J. Elast. 15, 117–124, 1985.
R. Russo, On the stability of solutions to the system of classical elastodynamics. Z. angew. Math. Phys. 37, 124–133, 1986.
R. Russo, Continuous data dependence and uniqueness in homogeneous linear elastostatics. Boll. Un. Mat. Ital. (6) 5-A, 227–234, 1986.
R. Russo, An extension of the basic theorems of linear elastostatics to exterior domains. Ann. Univ. Ferrara 34, 101–119, 1988.
L. T. Wheeler, Some results in the linear dynamical theory of anisotropic elastic solids. Quart. Appl. Math. 38, 91–101, 1970.
L. T. Wheeler, On the uniqueness of solutions to the displacement problem in linear elastodynamics. J. Elast. 1, 121–124, 1971.
L. T. Wheeler & E. Sternberg, Some theorems in classical elastodynamics. Arch. Rational Mech. Anal. 31, 51–90, 1968.
C. A. Wilcox, Initial boundary value problems for linear hyperbolic partial differential equations of second-order. Arch. Rational Mech. Anal. 10, 361–400, 1962.
C. A. Wilcox, The domain of dependence inequality for symmetric hyperbolic systems, Bull. Amer. Math. Soc. 70, 149–154, 1964.
C. A. Wilcox, Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics. J. Elast. 9, 221–243, 1979.
Author information
Authors and Affiliations
Additional information
Communicated by K. R. Rajagopal
Rights 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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00375226