Abstract
An electrical potential U on a bordered real surface X in ℝ3 with isotropic conductivity function σ>0 satisfies the equation d(σ d c U)| X =0, where \(d^{c}= i(\bar{ \partial }-\partial )\), \(d=\bar{ \partial }+\partial \) are real operators associated with a complex (conformal) structure on X induced by the Euclidean metric of ℝ3. This paper gives an exact reconstruction of the conductivity function σ on X from the Dirichlet-to-Neumann mapping U| bX →σ d c U| bX . This paper extends to the case of Riemann surfaces the reconstruction schemes of R. Novikov (Funkt. Anal. Prilozh. 22(4):11–22, 1988) and of A. Bukhgeim (J. Inv. Ill-posed Probl. 16:19–34, 2008), given for the case X⊂ℝ2. The paper extends and corrects the statements of Henkin and Michel (J. Geom. Anal. 18:1033–1052, 2008), where the inverse boundary value problem on the Riemann surfaces was first considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beals, R., Coifman, R.: Multidimensional Inverse Scattering and Nonlinear Partial Differential Equations. Proc. Symp. Pure Math., vol. 43, pp. 45–70. AMS Providence (1985)
Beals, R., Coifman, R.: The spectral problem for the Davey–Stewartson and Ishimori hierarchies. In: Nonlinear Evolution Equations: Integrability and Spectral Methods. Proc. Workshop, Como, Italy 1988, Proc. Nonlinear Sci., pp. 15–23 (1990)
Berenstein, C., Dostal, M.: Some remarks on convolution equations. Ann. Inst. Fourier 23, 55–73 (1973)
Boiti, M., Leon, J., Manna, M., Pempinelli, F.: On a spectral transform of a KDV-like equation related to the Schrödinger operator in the plane. Inverse Probl. 3, 25–36 (1987)
Bukhgeim, A.L.: Recovering a potential from the Cauchy data in the two-dimensional case. J. Inv. Ill-posed Probl. 16, 19–34 (2008)
Calderon, A.P.: On an inverse boundary problem. In: Seminar on Numerical Analysis and Its Applications to Continuum Physics, pp. 61–73. Soc. Brasiliera de Matematica, Rio de Janeiro (1980)
Druskin, V.L.: The unique solution of the inverse problem in electrical surveying and electrical well logging for piecewise-constant conductivity. Phys. Solid Earth 18(1), 51–53 (1982)
Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: The Schrödinger equation in a periodic field and Riemann surfaces. Dokl. Akad. Nauk SSSR 229, 15–18 (1976) (in Russian), Sov. Math. Dokl. 17, 947–951 (1976)
Ehrenpreis, L.: Solutions of some problems of division II. Am. J. Math. 77, 286–292 (1955)
Faddeev, L.D.: Increasing solutions of the Schrödinger equation. Dokl. Akad. Nauk SSSR 165, 514–517 (1965) (in Russian), Sov. Phys. Dokl. 10, 1033–1035 (1966)
Faddeev, L.D.: The inverse problem in the quantum theory of scattering II. Curr. Probl. Math. 3, 93–180 (1974) (in Russian), J. Sov. Math. 5, 334–396 (1976)
Fedorjuk, M.V.: Asymptotic: Integrals and Series. Nauka, Moscow (1987) (in Russian)
Gelfand, I.M.: Some problems of functional analysis and algebra. In: Proc. Int. Congr. Math., Amsterdam, pp. 253–276 (1954)
Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)
Grinevich, P.G., Novikov, S.P.: Two-dimensional inverse scattering problem for negative energies and generalized analytic functions. Funkt. Anal. Prilozh. 22(1), 23–33 (1988) (in Russian), Funct. Anal. and Appl. 22, 19–27 (1988)
Guillarmou, C., Tzou, L.: Calderón inverse problem for the Schrödinger operator on Riemann surfaces, arXiv:0904.3804 (2009) v.1
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
Henkin, G.M.: Cauchy–Pompeiu type formulas for \(\bar{ \partial }\) on affine algebraic Riemann surfaces and some applications, arXiv:0804.3761 (2008) v.1, (2010) v.2
Henkin, G.M., Polyakov, P.L.: Homotopy formulas for the \(\bar{\partial}\)-operator on ℂP n and the Radon–Penrose transform. Math. USSR Izv. 28, 555–587 (1987)
Henkin, G., Michel, V.: On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator, GAFA. Geom. Funct. Anal. 17, 116–155 (2007)
Henkin, G., Michel, V.: Inverse conductivity problem on Riemann surfaces. J. Geom. Anal. 18, 1033–1052 (2008)
Hodge, W.: The Theory and Applications of Harmonic Integrals. Cambridge Univ. Press, Cambridge (1952)
Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1990)
Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements II. Commun. Pure Appl. Math. 38, 644–667 (1985)
Nachman, A.: Global uniqueness for a two-dimensional inverse boundary problem. Ann. Math. 143, 71–96 (1996)
Novikov, R.: Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy. Funkt. Anal. Prilozh. 20(3), 90–91 (1986) (in Russian), Funct. Anal. Appl. 20, 246–248 (1986)
Novikov, R.: Multidimensional inverse spectral problem for the equation −Δψ+(v(x)−Eu(x))ψ=0. Funkt. Anal. Prilozh. 22(4), 11–22 (1988) (in Russian), Funct. Anal. Appl. 22, 263–278 (1988)
Novikov, R.: The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator. J. Funct. Anal. 103(2), 409–463 (1992)
Novikov, S.P., Veselov, A.P.: Two-dimensional Schrödinger operators in periodic fields. Curr. Probl. Math. 23, 3–32 (1983) (in Russian)
Rodin, Y.: Generalized Analytic Functions on Riemann Surfaces. Lecture Notes Math., vol. 1288. Springer, Berlin (1987)
Sylvester, J., Uhlmann, G.: A uniqueness theorem for an inverse boundary value problem in electrical prospection. Commun. Pure Appl. Math. 39, 91–112 (1986)
Tsai, T.Y.: The Schrödinger operator in the plane. Inverse Probl. 9, 763–787 (1993)
Vekua, I.N.: Generalized Analytic Functions. Pergamon, Elmsford (1962)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Henkin, G.M., Novikov, R.G. On the Reconstruction of Conductivity of a Bordered Two-dimensional Surface in ℝ3 from Electrical Current Measurements on Its Boundary. J Geom Anal 21, 543–587 (2011). https://doi.org/10.1007/s12220-010-9158-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-010-9158-8