An inverse problem and an observability inequality for the Lamé system with stress boundary condition

Abstract

In this article, by a Carleman estimate for solutions of the two-dimensional non-stationary Lamé system with the stress boundary condition, we obtain the conditional stability estimate in determining the density and two Lamé coefficients which vary spatially by a single observation in a neighbourhood of a suitable subboundary. Moreover, we prove an observability inequality with data measured in the same subdomain.

Keywords:

• Carleman estimate
• Lamé system
• stress boundary condition
• inverse problems
• observability

2000 AMS Subject Classifications::

• 35R30
• 35B60
• 74B05

Acknowledgements

Most of this article has been written during the stays of the first author at Graduate School of Mathematical Sciences of the University of Tokyo in 2007 and 2008. The author thanks the school for the hospitality. The first author was supported partially by the NSF Grant DMS-0808130. Both the authors thank the anonymous referee for invaluable comments.