Carleman estimates and inverse problems for Dirac operators

Salo, Mikko; Tzou, Leo

doi:10.1007/s00208-008-0301-9

Cited by 63 publications

(92 citation statements)

References 32 publications

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“…In [11], the authors give a result of the Carleman estimates for Dirac operators. The proof uses the fact that Dirac squared is the Laplacian.…”

Section: Carleman Estimates With Polynomial Weightsmentioning

confidence: 99%

“…The proof uses the fact that Dirac squared is the Laplacian. Inspired by [6,11], one can find a way to obtain the Carleman estimates with polynomial weights for Lamé operator directly from a corresponding estimates for the Laplacian in three dimensions, since the Lamé operator is a composition of two first order operators, one of which is a Dirac operator. For doing this, we need to use the semiclassical Sobolev spaces…”

Section: Carleman Estimates With Polynomial Weightsmentioning

confidence: 99%

“…A(∂) is a Dirac operator since A 2 = −ΔI 4 . We refer [11] for Carleman estimates for general Dirac operators. Now we turn to the Lamé system (1.1).…”

Section: Carleman Estimates With Polynomial Weightsmentioning

confidence: 99%

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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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“…In [11], the authors give a result of the Carleman estimates for Dirac operators. The proof uses the fact that Dirac squared is the Laplacian.…”

Section: Carleman Estimates With Polynomial Weightsmentioning

confidence: 99%

Section: Carleman Estimates With Polynomial Weightsmentioning

confidence: 99%

See 1 more Smart Citation

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

2010

ESAIM: COCV

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“…[14,20,26] and in the 3-dimensional case, see [23,28,29,53,59]. In the 3-dimensional case all these papers deal with perturbations of the canonical Dirac operator D 0 in R 3 or Ω ⊂ R 3 ,…”

Section: Introductionmentioning

confidence: 99%

Inverse problems and index formulae for Dirac operators

Kurylev

Lassas

2009

Advances in Mathematics

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We consider a Dirac-type operator D P on a vector bundle V over a compact Riemannian manifold (M, g) with a non-empty boundary. The operator D P is specified by a boundary condition P (u| ∂M ) = 0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L 2 (M, V ) into two orthogonal subspaces X + ⊕ X − . Under certain conditions, the operator D P restricted to X + and X − defines a pair of Fredholm operators which maps X + → X − and X − → X + correspondingly, giving rise to a superstructure on V . In this paper we consider the questions of determining the index of D P and the reconstruction of (M, g), V and D P from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M × R + of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of D P . We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M × C 4 , M ⊂ R 3 .

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“…A Carleman estimate approach for a different first order system, related to the Pauli Dirac operator, was presented in [ST09] where the method involved decoupling the Pauli Dirac operator into a second order differential operator with the Laplacian as its principal part. A suitable Carleman estimate was then applied to the decoupled equation to recover the coefficients.…”

Section: Introductionmentioning

confidence: 99%

Partial data inverse problems for Maxwell equations via Carleman estimates

Chung

Ola

Salo

et al. 2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. In this article we consider an inverse boundary value problem for the timeharmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of BukhgeimUhlmann and Kenig-Sjöstrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.

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Carleman estimates and inverse problems for Dirac operators

Cited by 63 publications

References 32 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

Inverse problems and index formulae for Dirac operators

Partial data inverse problems for Maxwell equations via Carleman estimates

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