The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system

Abstract

Abstract. In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous anisotropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.

Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems

“…Another application of the UCP to the limiting absorption principle for the same system was considered by the authors in [19]. To study our inverse problems here, the Runge approximation property with Dirichlet constraints for (1.2) is a key ingredient.…”

Reconstruction of cracks in an inhomogeneous anisotropic medium using point sources

“…Having obtained the uniqueness result, we can use the same method in [18] to prove the limiting absorption principle for the operator L in R n . In [18], the limiting absorption principle for L with Lipschitz elastic tensor in R 2 were established.…”

“…In [18], the limiting absorption principle for L with Lipschitz elastic tensor in R 2 were established. The method can be extended to R 3 without essential modifications.…”

“…In Section 2, we will describe the radiation conditions derived in Natroshvili's papers [19] and [20]. In Section 3, we prove the limiting absorption principle for the inhomogeneous anisotropic elasticity system in R n based on the recent work [18]. Then the limiting absorption principle will be used to establish the existence of the scattering solution in the whole space.…”

On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem

“…for r 2 R 2 n G satisfies the coupled system of elasticity equations, propagates outwardly and has zero farfield pattern (Nakamura and Wang, 2004;Natroshvili, 1996). From the generalized RellichÕs lemma for anisotropic elasticity (Nakamura and Wang, 2004), we conclude that (U 1 , U 2 ) = (0, 0) in R 2 n G. With the aid of the definitions (99) and (100), one can show that this also implies w = 0 in R 2 n G, apart indifferent rigid motions.…”