Boundary integral equation method in the steady state oscillation problems for ansiotropic bodies (Q2785652)

The steady state oscillation of an anisotropic elastic body is studied in the paper under the limiting absorbtion principle and where the fundamental matrices maximally decay at infinity, assuring the existence of the solution. The basic homogeneous equations of steady state oscillations for anisotropic bodies have the form (matricial) NEWLINE\[NEWLINEA(D,\omega)u(x)\equiv A(D)u(x)+\omega^2u(x)=0,\quad x\in\Omega,NEWLINE\]NEWLINE where \(u=(u^1,u^2,u^3)\) is the complex displacement vector, \(\omega>0\) an oscillation parameter, \(\Omega\subset\mathbb{R}^3\) be bounded domain with smooth boundary \(\partial\Omega= S\), \(D=\nabla=(D_1,D_2,D_3)\), \(D_k=\partial/\partial x_k\), \(A(\xi)=|A_{kp}(\xi)|_{3\times 3}\), \(A_{kp}(\xi)= c_{kjpq}\xi_j\xi_q\), \(k,p=1,2,3\), \(c_{kjpq}= c_{pqkj}= c_{jkpq}\) are elastic constants. This system of equations is supplemented by some boundary conditions for \(x\in S\).NEWLINENEWLINENEWLINEFor this boundary value problem, the author finds the fundamental matrices and representation formulae, as usually in the linear case, and proves some existence and uniqueness theorems.