Three spheres inequalities for a two-dimensional elliptic system and its application

Abstract

In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system. We then give an application of these three spheres inequalities to the inverse problem of identifying cavities by partial boundary measurements.

Three spheres inequalities and unique continuation for a three-dimensional Lamé system of elasticity with C^1 coefficients

“…holds for u ∈ H 2 (Ω) being a solution of (1.1), where C and σ ∈ (0, 1) are two constants depending on R 1 R , R 2 R , λ C 1 (Ω) and µ C 1 (Ω) . Quantitative estimates like (1.2) have been shown to be extremely useful in the treatment of the unique continuation principle and the inverse boundary value problems [2], [12], [13], [15]. Another result of our paper is related to the strong unique continuation.…”

“…Inspired by [12] and [15], we give a proof of the three spheres inequality (Theorem 1.1) as follows.…”

“…Unfortunately, the method mentioned above can not be used when the Lamé coefficients are C 1 , because one cannot apply divergence to (1.1) to diagonalize the system in this case. Unlike the approach used in [1], C. -L. Lin et al derived the three spheres inequality for a 2-dimensional elliptic system with W 1,∞ coefficients by another type of reduction of the Lamé system and Carleman estimates [15]. It is carried out by using an auxiliary function ∂ x 1 u + T ∂ x 2 u with an appropriate matrix T .…”