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

Abstract

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

Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions

“…This inequality and the Carleman estimate established in Proposition 5.1 of [9] for the elliptic equation implies…”

“…Lemma 4.6 is proved using a standard argument (e.g., [17]) and is similar to one found in [9]. For the sake of completeness, we include the proof in the Appendix.…”

“…[24]), much less is known for systems of partial differential equations coupled through principal terms, as is the case in Lamé systems. Carleman estimates for the Lamé systems are obtained with the use of recently derived Carleman estimates due to Imanuvilov and Yamamoto [9][10][11]. (See Lems.…”

Stability of Diffusion Coefficients in an Inverse Problem for the Lotka-Volterra Competition System

“…Interestingly for the first time, Klibanov [25] obtained the stability estimate for diffusion equation in an unbounded domain by using Carleman estimates. Imanuvilov and Yamamoto [20] discussed the uniqueness and the stability in determining spatially varying density and two Lamé coefficients for the two dimensional isotropic nonstationary Lamé system by a single measurement of solution over ω × (0, T ), where the subdomain ω satisfies a non-trapping condition and T > 0 is large enough. Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation has been proved by Bellassoued and Yamamoto [3].…”

Uniqueness and Stability of Determining the Residual Stress by One Measurement

“…In Imanuvilov et al (2003), they used this estimate to study the problem of identifying the density and Lamé coefficients by two sets of data measured in a boundary layer and a Hölder-type stability estimate. The continuation of this work is in Imanuvilov and Yamamoto (2005).…”