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February  2019, 13(1): 149-157. doi: 10.3934/ipi.2019008

Note on Calderón's inverse problem for measurable conductivities

Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genova, Italy

Received  February 2018 Revised  June 2018 Published  December 2018

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation ${\rm{div}} (σ \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calderón problem in three and higher dimensions in the $L^\infty$ case.

Citation: Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems and Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM Journal on Mathematical Analysis, 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.

[3]

K. AstalaM. Lassas and L. Päivärinta, The borderlines of the invisibility and visibility for Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98.  doi: 10.2140/apde.2016.9.43.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

M. I. Belishev and A. F. Vakulenko, On algebras of harmonic quaternion fields in $\mathbb{R}^3$, arXiv preprint, arXiv: 1710.00577, 2017.

[6]

M. I. Belishev, On algebras of three-dimensional quaternion harmonic fields, Journal of Mathematical Sciences, 226 (2017), 701-710.  doi: 10.1007/s10958-017-3559-1.

[7]

M. I. Belishev, Some remarks on the impedance tomography problem for 3d-manifolds, Cubo, 7 (2005), 43-55. 

[8]

L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954.

[9]

F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, vol. 76, Pitman Books Limited, 1982.

[10]

R. M. Brown and R. H. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in Lp, p > 2n, The Journal of Fourier Analysis and Applications, 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[11]

A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics (Soc. Brasil. Mat., Rio de Janeiro), (1980), 65-73. 

[12]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, in Forum of Mathematics, Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9.

[13]

B. B. Delgado and R. M. Porter, General solution of the inhomogeneous div-curl system and consequences, Advances in Applied Clifford Algebras, 27 (2017), 3015-3037.  doi: 10.1007/s00006-017-0805-z.

[14]

B. B. Delgado and R. M. Porter, Hilbert transform for the three-dimensional vekua equation, arXiv preprint, arXiv: 1803.03293, 2018.

[15]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035. 

[16]

K. Gürlebeck and W. Sprößig, Quaternionic analysis and elliptic boundary value problems, Int. Series of Numerical Mathematics, 89.

[17]

B. Haberman, Uniqueness in Calderón problem for conductivities with unbounded gradient, Communications in Mathematical Physics, 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[18]

C. LiA. G. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Revista Matematica Iberoamericana, 10 (1994), 665-721.  doi: 10.4171/RMI/164.

[19]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.  doi: 10.2307/2118653.

[20]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169.  doi: 10.2307/1971291.

[21]

V. I. Vekua, Generalized Analytic Functions, Pergamon Press London, 1962.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM Journal on Mathematical Analysis, 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.

[3]

K. AstalaM. Lassas and L. Päivärinta, The borderlines of the invisibility and visibility for Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98.  doi: 10.2140/apde.2016.9.43.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

M. I. Belishev and A. F. Vakulenko, On algebras of harmonic quaternion fields in $\mathbb{R}^3$, arXiv preprint, arXiv: 1710.00577, 2017.

[6]

M. I. Belishev, On algebras of three-dimensional quaternion harmonic fields, Journal of Mathematical Sciences, 226 (2017), 701-710.  doi: 10.1007/s10958-017-3559-1.

[7]

M. I. Belishev, Some remarks on the impedance tomography problem for 3d-manifolds, Cubo, 7 (2005), 43-55. 

[8]

L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954.

[9]

F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, vol. 76, Pitman Books Limited, 1982.

[10]

R. M. Brown and R. H. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in Lp, p > 2n, The Journal of Fourier Analysis and Applications, 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[11]

A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics (Soc. Brasil. Mat., Rio de Janeiro), (1980), 65-73. 

[12]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, in Forum of Mathematics, Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9.

[13]

B. B. Delgado and R. M. Porter, General solution of the inhomogeneous div-curl system and consequences, Advances in Applied Clifford Algebras, 27 (2017), 3015-3037.  doi: 10.1007/s00006-017-0805-z.

[14]

B. B. Delgado and R. M. Porter, Hilbert transform for the three-dimensional vekua equation, arXiv preprint, arXiv: 1803.03293, 2018.

[15]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035. 

[16]

K. Gürlebeck and W. Sprößig, Quaternionic analysis and elliptic boundary value problems, Int. Series of Numerical Mathematics, 89.

[17]

B. Haberman, Uniqueness in Calderón problem for conductivities with unbounded gradient, Communications in Mathematical Physics, 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[18]

C. LiA. G. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Revista Matematica Iberoamericana, 10 (1994), 665-721.  doi: 10.4171/RMI/164.

[19]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.  doi: 10.2307/2118653.

[20]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169.  doi: 10.2307/1971291.

[21]

V. I. Vekua, Generalized Analytic Functions, Pergamon Press London, 1962.

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