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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

Figure 1.  Integration regions in equation (8)
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