August  2020, 14(4): 701-718. doi: 10.3934/ipi.2020032

Reconstruction of the derivative of the conductivity at the boundary

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E48009 Bilbao, Basque Country – Spain

Received  September 2019 Revised  February 2020 Published  May 2020

Fund Project: The author is supported by the Basque Government and by the Spanish State Research Agency

We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. The method of reconstruction works for isotropic conductivities with low regularity. This boundary determination for rough conductivities implies the uniqueness of the conductivity in the whole domain $ \Omega $ when it lies in $ W^{1+\frac{n-5}{2p}+, p}(\Omega) $, for dimensions $ n\ge 5 $ and for $ n\le p<\infty $.

Citation: Felipe Ponce-Vanegas. Reconstruction of the derivative of the conductivity at the boundary. Inverse Problems and Imaging, 2020, 14 (4) : 701-718. doi: 10.3934/ipi.2020032
References:
[1]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[2]

G. Alessandrini and R. Gaburro, The local Calderòn problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207-241.  doi: 10.1016/j.aam.2004.12.002.

[4]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: The full story, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1355-1376.  doi: 10.1016/j.anihpc.2017.11.007.

[5]

R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, J. Inverse Ill-Posed Probl., 9 (2001), 567-574.  doi: 10.1515/jiip.2001.9.6.567.

[6]

R. M. Brown and R. H. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^p, p>2n$, J. Fourier Anal. Appl., 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[7]

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

[8]

M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of $S^M$-valued functions, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 127-144. 

[9]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[10]

B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[11]

S. Ham, Y. Kwon and S. Lee, Uniqueness in the Calderón problem and bilinear restriction estimates, preprint, arXiv: 1903.09382.

[12]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.

[13]

J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math., 58 (1987), 47-65.  doi: 10.1007/BF01169082.

[14]

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

[15]

G. NakamuraS. SiltanenK. Tanuma and S. Wang, Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map, Computing, 75 (2005), 197-213.  doi: 10.1007/s00607-004-0095-x.

[16]

G. Nakamura and K. Tanuma, Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map, in Recent Development in Theories & Numerics (eds. Y.-C. Hon, M. Yamamoto, J. Cheng and J.-Y. Lee), World Sci. Publ., River Edge, NJ, 2003,192–201. doi: 10.1142/9789812704924_0017.

[17]

F. Ponce-Vanegas, The bilinear strategy for Calderón's problem, preprint, arXiv: 1908.04050.

[18]

E. SomersaloM. CheneyD. Isaacson and E. Isaacson, Layer stripping: A direct numerical method for impedance imaging, Inverse Problems, 7 (1991), 899-926.  doi: 10.1088/0266-5611/7/6/011.

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.

[20]

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

[21]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary—continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.

[22]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

show all references

References:
[1]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[2]

G. Alessandrini and R. Gaburro, The local Calderòn problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207-241.  doi: 10.1016/j.aam.2004.12.002.

[4]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: The full story, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1355-1376.  doi: 10.1016/j.anihpc.2017.11.007.

[5]

R. M. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: A pointwise result, J. Inverse Ill-Posed Probl., 9 (2001), 567-574.  doi: 10.1515/jiip.2001.9.6.567.

[6]

R. M. Brown and R. H. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^p, p>2n$, J. Fourier Anal. Appl., 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[7]

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

[8]

M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of $S^M$-valued functions, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 127-144. 

[9]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[10]

B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[11]

S. Ham, Y. Kwon and S. Lee, Uniqueness in the Calderón problem and bilinear restriction estimates, preprint, arXiv: 1903.09382.

[12]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.

[13]

J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math., 58 (1987), 47-65.  doi: 10.1007/BF01169082.

[14]

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

[15]

G. NakamuraS. SiltanenK. Tanuma and S. Wang, Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map, Computing, 75 (2005), 197-213.  doi: 10.1007/s00607-004-0095-x.

[16]

G. Nakamura and K. Tanuma, Formulas for reconstructing conductivity and its normal derivative at the boundary from the localized Dirichlet to Neumann map, in Recent Development in Theories & Numerics (eds. Y.-C. Hon, M. Yamamoto, J. Cheng and J.-Y. Lee), World Sci. Publ., River Edge, NJ, 2003,192–201. doi: 10.1142/9789812704924_0017.

[17]

F. Ponce-Vanegas, The bilinear strategy for Calderón's problem, preprint, arXiv: 1908.04050.

[18]

E. SomersaloM. CheneyD. Isaacson and E. Isaacson, Layer stripping: A direct numerical method for impedance imaging, Inverse Problems, 7 (1991), 899-926.  doi: 10.1088/0266-5611/7/6/011.

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.

[20]

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

[21]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary—continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219.  doi: 10.1002/cpa.3160410205.

[22]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

Figure 1.  Integration regions in equation (8)
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