Carleman estimate for a stationary isotropic Lamé system and the applications

Abstract

For the isotropic stationary Lamé system with variable coefficients equipped with the Dirichlet or surface stress boundary condition, we obtain a Carleman estimate such that (i) the right hand side is estimated in a weighted L ²-space and (ii) the estimate includes nonhomogeneous surface displacement or surface stress. Using this estimate we establish the conditional stability in Sobolev's norm of the displacement by means of measurements in an arbitrary subdomain or measurements of surface displacement and stress on an arbitrary subboundary. Finally by the Carleman estimate, we prove the uniqueness and conditional stability for an inverse problem of determining a source term by a single interior measurement.

Keywords:

• Lamé system
• Carleman estimate
• Unique continuation
• Inverse source problem
• 1991 Mathematics Subject Classifications: 35B60
• 35R30
• 74B05

Acknowledgements

Oleg Imanuvilov was supported in part by NSF Grant DMS 02-05148. Masahiro Yamamoto was supported in part by Grants 15340027 and 15654015 from the Japan Society for the Promotion of Science. The authors thank Professors Saburou Saitoh and Masaru Ikehata (Gunma University) for invaluable comments.

Notes

^†E-mail: [email protected]

Additional information

Notes on contributors

Masahiro Yamamoto †

†E-mail: [email protected]