Initial-boundary value problem for systems of equations describing wave motion in a fluid layer on an elastic base (Q1206252)

The title initial-boundary value problem is \(\nabla^ 2\psi= 0\), \(-H< z< 0\), \(\nabla^ 2 U+ (c^ 2- 1)\nabla\text{ div } U= \mu^ 2 U_{tt}\), \(z<- H\), \(\psi_{tt}+ \psi_ z= 0\), \(z= 0\), where \(\psi\) is the potential of the moving liquid \((-H\leq z\leq 0)\) and \(U\) is the displacement vector in the elastic half-space \(z\leq -H\). A sketch of the proof of the following statement is given: the system in question can be reduced to the form \(\partial^ 2 W/\partial t^ 2={\mathcal L}W\), where \(\mathcal L\) is a certain essentially self-adjoint operator generated by an explicitly written linear system of equations. The spectrum of \(\widehat{\mathcal L}\) consists of discrete and continuous parts, and the corresponding eigenfunctions form a complete orthonormal system with respect to a certain scalar product.