Unique continuation for systems with Lamé principal part

Weck, Norbert

doi:10.1002/mma.231

Cited by 25 publications

(23 citation statements)

References 10 publications

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“…They proved the Carleman estimate by pseudodifferential calculus. Then Ang et al gave a result for λ ∈ C 2 (R n ), μ ∈ C 3 (R n ) [2]; Weck proved a result for λ, μ ∈ C 2 (R n ) [12,13]. On the other hand, the result on the strong unique continuation (SUCP) for the Lamé system was first obtained by Alessandrini and Morassi for n ≥ 2, λ, μ ∈ C 1,1 (R n ) [1].…”

Section: Introductionmentioning

confidence: 98%

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

2010

ESAIM: COCV

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Abstract. This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, μ in three dimensions, div(μ(∇u + ∇u t )) + ∇(λdivu) + V u = 0 where λ and μ are Lipschitz continuous and V ∈ L ∞ . The method is based on the Carleman estimate with polynomial weights for the Lamé operator.Mathematics Subject Classification. 35B60, 74B05.

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Section: Introductionmentioning

confidence: 98%

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

2010

ESAIM: COCV

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“…Besides the radiation conditions, one of the key ingredients in [2] and [10] is the unique continuation property for the isotropic elasticity system. The unique continuation property for this system has been proved by several authors; see for example [1], [3], [17], and [18]. In contrast to the isotropic setup, the unique continuation property for the general anisotropic elasticity system still poses a challenging open problem.…”

Section: Introductionmentioning

confidence: 99%

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

Nakamura

Wang

2004

Trans. Amer. Math. Soc.

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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.

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“…Instead, we only mention some related results on the elasticity system. When the medium is isotropic, the UCP has been established in [1], [4] and [21]. Moreover, a strong unique continuation property for the isotropic system was recently proven in [2] and in [15].…”

Section: Introductionmentioning

confidence: 99%

“…Our proof of the UCP for (1.2) via (2.11) relies on some delicate Carleman estimates. To deal with the lower triangular matrix function N , we will borrow some ideas from [21] (or [18]). The proof of the UCP is given in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, as a by-product of our UCP result, we prove that the UCP holds for the two-dimensional isotropic elasticity system with locally Lipschitz Lamé coefficients, which is an improvement of previous UCP results for the same system in two dimensions. We remark that for the UCP result obtained in [1] or [21], the Lamé coefficients are required to be at least C 2 , while for the strong UCP proved in [2], one needs C 1,1 coefficients. Our study of the UCP for the inhomogeneous two-dimensional anisotropic elasticity system (1.2) is motivated by its application to inverse problems.…”

Section: Introductionmentioning

confidence: 99%

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

Nakamura

Wang

2006

Trans. Amer. Math. Soc.

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Abstract. Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.

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Unique continuation for systems with Lamé principal part

Cited by 25 publications

References 10 publications

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

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

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

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