Solvability of singular integral system connected with seismic waves around the borehole

We study in this article a boundary-value problem arising in the propagation of waves in an elastic half-space covered by a layer with a vertical borehole. We first show a uniqueness theorem under some restrictions on the solution. For the existence, we use the direct integral equation method. We obtain a singular integral system on the half-line. For the solvability, we reduce this system to an elliptic pseudodifferential equation and establish the Fredholm property. Finally, we compute the index of the associated operator for various values of Poissonʹs ratio.