## Global uniqueness from partial Cauchy data in two dimensions. Arxiv preprint arXiv:0810.2286 (2008)

Citations: | 12 - 1 self |

### BibTeX

@MISC{Imanuvilov08globaluniqueness,

author = {Oleg Yu. Imanuvilov and Gunther Uhlmann and Masahiro Yamamoto},

title = {Global uniqueness from partial Cauchy data in two dimensions. Arxiv preprint arXiv:0810.2286},

year = {2008}

}

### OpenURL

### Abstract

Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.

### Citations

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Analysis of linear partial differential operators
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Citation Context ...2 (Ω) and g|H = 0, then (2.10) |RΦ,τg(x)| + | ˜ RΦ,τg(x)| ≤ C/τ 2 for all x ∈ Oǫ/2. The proof uses the Cauchy-Riemann equations and stationary phase (e.g., Section 4.5.3 in [13], Chapter VII, §7.7 in =-=[16]-=-). See also the proof of Proposition 3.4 in [17]. Denote r(z) = Π ℓ k=1 (z − ˜zk) where H = {˜x1, . . ., ˜xℓ}, ˜zk = ˜x1,k + i˜x2,k. The following proposition can be proved similarly to Proposition 3.... |

762 | Partial differential equations
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Citation Context ...for all x ∈ Oǫ/2. If g ∈ C 2 (Ω) and g|H = 0, then (2.10) |RΦ,τg(x)| + | ˜ RΦ,τg(x)| ≤ C/τ 2 for all x ∈ Oǫ/2. The proof uses the Cauchy-Riemann equations and stationary phase (e.g., Section 4.5.3 in =-=[13]-=-, Chapter VII, §7.7 in [16]). See also the proof of Proposition 3.4 in [17]. Denote r(z) = Π ℓ k=1 (z − ˜zk) where H = {˜x1, . . ., ˜xℓ}, ˜zk = ˜x1,k + i˜x2,k. The following proposition can be proved ... |

229 |
A global uniqueness theorem for an inverse boundary value problem
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(Show Context)
Citation Context ... boundary of the body. This problem was proposed by Calderón [9] and is also known as Calderón’s problem. In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in =-=[28]-=-. In [25], [5] the global uniqueness result was extended to less regular conductivities. Also see [14] as for the determination of more singular conormal conductivities. In dimension n ≥ 3 global uniq... |

151 |
Global uniqueness for a two-dimensional inverse boundary value problem
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Citation Context ...quation with bounded potentials in [28]. The case of more singular conormal potentials was studied in [14]. In two dimensions the first global uniqueness result for Calderón’s problem was obtained in =-=[24]-=- for C 2 -conductivities. Later the regularity assumptions were relaxed in [6], and [2]. In particular, the paper [2] proves uniqueness for L ∞ - conductivities. In two dimensions a recent result of B... |

113 |
Analysis of Toeplitz Operators
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Citation Context ...(z)q2) − M2(z) ∂zΦ (q1a) − M1(z) ∂zΦ + qa ∂−1 z + qa ∂−1 z (q2a(z)) ) − M4(z) dx ∂zΦ (q1a) − M3(z) ∂zΦ ) dx = o(1). as τ → +∞. Passing to the limit in this equality and applying Bohr’s theorem (e.g., =-=[4]-=-, p.393), we finish the proof of the proposition. □ ( ) 1 . τPARTIAL DIRICHLET-TO-NEUMANN MAP 15 We need the following proposition in the construction of the phase function Φ. Let ˜y1, . . ., ˜ym ∈ Ω... |

76 |
Calderón’s inverse conductivity problem in the plane
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Citation Context ...as studied in [14]. In two dimensions the first global uniqueness result for Calderón’s problem was obtained in [24] for C 2 -conductivities. Later the regularity assumptions were relaxed in [6], and =-=[2]-=-. In particular, the paper [2] proves uniqueness for L ∞ - conductivities. In two dimensions a recent result of Bukgheim [7] gives unique identifiability of the potential from Cauchy data measured on ... |

60 | The Calderón Problem with partial data
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Citation Context ... was relaxed to C 3/2+α with some α > 0. The corresponding stability estimates are proved in [15]. As for the stability estimates for the magnetic Schrödinger equation with partial data, see [29]. In =-=[20]-=-, the result in [8] was generalized to show that by all possible pairs of Dirichlet data on an arbitrary open subset Γ+ of the boundary and Neumann data on a slightly larger open domain than ∂Ω \ Γ+, ... |

56 | An anisotropic inverse boundary value problem
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Citation Context ...s in [24], Lipschitz conductivities in [26] and merely L ∞ conductivities in [3]. The method of proof in all these papers is the reduction to the isotropic case performed using isothermal coordinates =-=[27]-=-. Using the same method and Corollary 1.1 we obtain the following result Theorem 1.2. Let σk = {σ ij k } ∈ C3+α (Ω) for k = 1, 2 and some positive α. Suppose that σk are positive definite symmetric ma... |

50 |
On an inverse boundary value problem
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Citation Context ...blem. In this inverse problem one attempts to determine the electrical conductivity of a body by measurements of voltage and current on the boundary of the body. This problem was proposed by Calderón =-=[9]-=- and is also known as Calderón’s problem. In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in [28]. In [25], [5] the global uniqueness result was extended to ... |

33 | Calderón’s inverse problem for anisotropic conductivity in the plane
- Astala, Lassas, et al.
(Show Context)
Citation Context ...he question of whether one can determine the conductivity up to the obstruction (1.6) has been solved for C 2 conductivities in [24], Lipschitz conductivities in [26] and merely L ∞ conductivities in =-=[3]-=-. The method of proof in all these papers is the reduction to the isotropic case performed using isothermal coordinates [27]. Using the same method and Corollary 1.1 we obtain the following result The... |

32 |
Recovering a Potential from Partial Cauchy Data
- Bukhgeim, Uhlmann
(Show Context)
Citation Context ...subset of the boundary, even for smooth potentials or conductivities. In dimension n ≥ 3 Isakov [18] proved global uniqueness assuming that Γ0 is a subset of a plane or a sphere. In dimensions n ≥ 3, =-=[8]-=- proves global uniqueness in determining a bounded potential for the Schrödinger equation with Dirichlet data supported on the whole boundary and Neumann data measured in roughly half the boundary. Th... |

31 |
Recovering a Potential from Cauchy data in the TwoDimensional Case
- Bukhgeim
(Show Context)
Citation Context ... -conductivities. Later the regularity assumptions were relaxed in [6], and [2]. In particular, the paper [2] proves uniqueness for L ∞ - conductivities. In two dimensions a recent result of Bukgheim =-=[7]-=- gives unique identifiability of the potential from Cauchy data measured on the whole boundary for the associated Schrödinger equation. As for the uniqueness in determining two coefficients, see [10],... |

27 |
Determining the magnetic Schrodinger operator from partial Cauchy data, preprint
- Ferreira, Kenig, et al.
(Show Context)
Citation Context ...domain than ∂Ω \ Γ+, one can uniquely determine the potential. The method of the proof uses Carleman estimates with non-linear weights. The case of the magnetic Schrödinger equation was considered in =-=[11]-=- and an improvement on the regularity of the coefficients is done in [22]. Stability estimates for the magnetic Schrödinger equation with partial data were proven in [29]. In two dimensions the first ... |

27 | Limiting carleman weights and anisotropic inverse problems, Invent - Ferreira, Kenig, et al. |

21 |
Vogelius M: ‘Identification of an unknown conductivity by means of measurements at the boundary
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- 1984
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Citation Context ...j ∂u (σ ) = 0 in Ω, ∂xi ∂xj i,j=1 u|∂Ω = g. The Dirichlet-to-Neumann map is defined by 2∑ Λσ(g) = i,j=1 It has been known for a long time that Λσ does not determine σ uniquely in the anisotropic case =-=[23]-=-. Let F : Ω → Ω be a diffeomorphism such that F(x) = x for and x from ∂Ω. Then where σ ij νi ΛF∗σ = Λσ, ∂u ∂xj |∂Ω. (1.6) F∗σ = (DF) · σ · (DF)T · F −1 . |detDF | Here DF denotes the differential of F... |

21 |
Generalized Analytic Functions, Pergamon Press
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Citation Context ...n 2.2 in [17] and Proposition 5.2 in appendix. Let us introduce the operators: ∂ −1 ∫ ∫ 1 g(ζ, ζ) g(ζ, ζ) z g = dζ ∧ dζ = −1 2πi Ω ζ − z π Ω ζ − z dξ2dξ1, ∂ −1 ∫ 1 z g = − 2πi Ω See e.g., pp.28-31 in =-=[31]-=- where ∂ −1 z know (e.g., p.47 and p.56 in [31]): ∫ g(ζ, ζ) g(ζ, ζ) dζ ∧ dζ = −1 ζ − z π Ω ζ − z dξ2dξ1 = ∂ −1 z g. and ∂ −1 z are denoted by T and T respectively. Then we Proposition 2.2. A) Let m ≥ ... |

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Citation Context ...em. In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in [28]. In [25], [5] the global uniqueness result was extended to less regular conductivities. Also see =-=[14]-=- as for the determination of more singular conormal conductivities. In dimension n ≥ 3 global uniqueness was shown for the Schrödinger equation with bounded potentials in [28]. The case of more singul... |

16 | Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in Lp, p > 2n
- Brown, Torres
- 2002
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Citation Context ...e body. This problem was proposed by Calderón [9] and is also known as Calderón’s problem. In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in [28]. In [25], =-=[5]-=- the global uniqueness result was extended to less regular conductivities. Also see [14] as for the determination of more singular conormal conductivities. In dimension n ≥ 3 global uniqueness was sho... |

14 | Determining nonsmooth first order terms from partial boundary measurements
- Knudsen, Salo
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Citation Context ... of the proof uses Carleman estimates with non-linear weights. The case of the magnetic Schrödinger equation was considered in [11] and an improvement on the regularity of the coefficients is done in =-=[22]-=-. Stability estimates for the magnetic Schrödinger equation with partial data were proven in [29]. In two dimensions the first general result was given by the authors in [17]. It is shown that the sam... |

12 |
Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems
- Heck, Wang
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Citation Context ...or the conductivity equation with C 2 conductivities. In [21] the regularity assumption on the conductivity was relaxed to C 3/2+α with some α > 0. The corresponding stability estimates are proved in =-=[15]-=-. As for the stability estimates for the magnetic Schrödinger equation with partial data, see [29]. In [20], the result in [8] was generalized to show that by all possible pairs of Dirichlet data on a... |

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11 |
Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions
- Brown, Uhlmann
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Citation Context ...entials was studied in [14]. In two dimensions the first global uniqueness result for Calderón’s problem was obtained in [24] for C 2 -conductivities. Later the regularity assumptions were relaxed in =-=[6]-=-, and [2]. In particular, the paper [2] proves uniqueness for L ∞ - conductivities. In two dimensions a recent result of Bukgheim [7] gives unique identifiability of the potential from Cauchy data mea... |

10 | The Calderón problem with partial data for less smooth conductivities
- Knudsen
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Citation Context ...a measured in roughly half the boundary. The proof relies on a Carleman estimate with a linear weight function. This implies a similar result for the conductivity equation with C 2 conductivities. In =-=[21]-=- the regularity assumption on the conductivity was relaxed to C 3/2+α with some α > 0. The corresponding stability estimates are proved in [15]. As for the stability estimates for the magnetic Schrödi... |

9 |
Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case
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Citation Context ...m [7] gives unique identifiability of the potential from Cauchy data measured on the whole boundary for the associated Schrödinger equation. As for the uniqueness in determining two coefficients, see =-=[10]-=-, [19]. In all the above mentioned articles, the measurements are made on the whole boundary. The purpose of this paper is to show the global uniqueness in two dimensions, both for the Schrödinger and... |

9 | Stability estimates for coefficients of magnetic Schrödinger equation from full and partial measurements
- Tzou
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Citation Context ...ductivity was relaxed to C 3/2+α with some α > 0. The corresponding stability estimates are proved in [15]. As for the stability estimates for the magnetic Schrödinger equation with partial data, see =-=[29]-=-. In [20], the result in [8] was generalized to show that by all possible pairs of Dirichlet data on an arbitrary open subset Γ+ of the boundary and Neumann data on a slightly larger open domain than ... |

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Complex geometrical optics for Lipschitz conductivities
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Citation Context ... of the body. This problem was proposed by Calderón [9] and is also known as Calderón’s problem. In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in [28]. In =-=[25]-=-, [5] the global uniqueness result was extended to less regular conductivities. Also see [14] as for the determination of more singular conormal conductivities. In dimension n ≥ 3 global uniqueness wa... |

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Anisotropic inverse problems
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Citation Context ...tion in (1.6) is matrix composition. The question of whether one can determine the conductivity up to the obstruction (1.6) has been solved for C 2 conductivities in [24], Lipschitz conductivities in =-=[26]-=- and merely L ∞ conductivities in [3]. The method of proof in all these papers is the reduction to the isotropic case performed using isothermal coordinates [27]. Using the same method and Corollary 1... |

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Citation Context ...e coefficients is done in [22]. Stability estimates for the magnetic Schrödinger equation with partial data were proven in [29]. In two dimensions the first general result was given by the authors in =-=[17]-=-. It is shown that the same global uniqueness result as [20] holds in this case. The two dimensional case has special features since one can construct a much larger set of complex geometrical optics s... |

5 |
Inverse problems for the Pauli Hamiltonian in two dimensions
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Citation Context ...gives unique identifiability of the potential from Cauchy data measured on the whole boundary for the associated Schrödinger equation. As for the uniqueness in determining two coefficients, see [10], =-=[19]-=-. In all the above mentioned articles, the measurements are made on the whole boundary. The purpose of this paper is to show the global uniqueness in two dimensions, both for the Schrödinger and condu... |