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Energy inequalities and the domain of influence theorem in classical elastodynamics. (English) Zbl 0547.73011
The authors suggest a generalization of the concept of domain of influence for the Cauchy problem of classical anisotropic elastodynamics when the density $$\rho$$ and elastic tensor C are such that $$\rho^{-1}C$$ is not bounded and C is symmetric but only positive semi-definite. The governing equations are $$\rho\ddot u=\nabla\cdot C[\nabla u]+b$$ in $$\Omega \times]0,T[$$; $$u=u^*$$ on $$\partial_ 1\Omega\times]0,T[$$; $$C[\nabla u]\cdot\nu =s^*$$ on $$\partial_ 2\Omega\times]0,T[$$; $$u=u_ 0$$ on $$\Omega \times (0)$$; $$\dot u=\dot u_ 0$$ on $$\Omega \times (0)$$. Theorems are established under the assumption that $$|\rho^{- 1}C| =O(r^{2+\epsilon})$$, $$\forall\epsilon >0$$ as $$r\to\infty$$. A number of a priori estimates are derived along the way.
Reviewer: L.E.Payne

##### MSC:
 74E10 Anisotropy in solid mechanics 74J99 Waves in solid mechanics 35G15 Boundary value problems for linear higher-order PDEs 35L55 Higher-order hyperbolic systems
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