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A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: a priori estimates in \(H^1\). (English) Zbl 1146.35323

Summary: We provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain \(\Omega \subset \mathbb R^3\) (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 \(H^1\) 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.

MSC:

35B45 A priori estimates in context of PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
74J05 Linear waves in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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