A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in H1 ✩

In this note, we provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain Ω ⊂ R3 (the scatterer). The boundary condition at infinity is given by the Kupradze– Sommerfeld radiation condition and involves different Sommerfeld conditions on different components of the field. Our results are obtained by setting up the problem as a variational problem in the Sobolev space H1 on a bounded domain. We use a nonlocal boundary condition which is related to the Dirichlet to Neumann conditions used for acoustic and electromagnetic scattering problems. We obtain stability results for the source problem, a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences