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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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