Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems

Abstract

A method based upon the Lagrange identity has been used by Brun [2] in the linear theories of thermoelasticity and viscoelasticity to establish uniqueness of the solution to the initial boundary value problem on bounded three-dimensional regions. A major feature of Brun's analysis is that it does not require any sign-definiteness assumptions on, for instance, the elasticities. The technique was extended by Knops and Payne [14] to derive certain continuous dependence results in linear elastodynamics, again for a bounded region. These authors had earlier recovered Brun's uniqueness result for linear elasticity [11] and derived other continuous dependence results based upon logarithmic convexity arguments [12, 14] (see also [13] for a similar treatment of thermoelasticity). Levine [18] later treated an abstract version of the Brun approach and applied it to a family of abstract linear operator equations. Among his results is a simplified proof that equipartition of the kinetic and potential energies occurs. Other applications of the Lagrange identity in proofs of uniqueness for bounded regions include those by Naghdi and Trapp [19] for a Cosserat surface, and by Green [9] for a theory of linear thermoelasticity that allows second sound.