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In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lame system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lame coefficients by a single measurement of solution over (0,T ) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

Carleman estimates for the non-stationary Lamé system and the application to an inverse problem