pith. sign in

arxiv: 2001.05155 · v5 · submitted 2020-01-15 · 🧮 math.AP

Reconstruction of Rough Conductivities from Boundary Measurements

Ashwin Tarikere This is my paper

Pith reviewed 2026-05-24 15:32 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse conductivity problemDirichlet-to-Neumann mapNachman's procedureSobolev regularityCalderón problemstability estimatesboundary measurements
0
0 comments X

The pith

Nachman's procedure reconstructs conductivities in H^{3/2,2n} from the Dirichlet-to-Neumann map when the conductivity equals one near the boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Nachman's 1988 reconstruction method remains valid for recovering a conductivity function from its Dirichlet-to-Neumann map when the conductivity lies in the Sobolev space H^{3/2,2n} inside a smooth bounded domain in dimensions three and higher, provided it equals one near the boundary. This lowers the regularity threshold from prior results that demanded more smoothness. A sympathetic reader would care because it enlarges the class of conductivities for which unique recovery from boundary voltage and current data is guaranteed. The work further derives a logarithmic stability estimate for the inverse problem under a marginally stronger regularity condition in the space H^{2-s,n/s} for parameters between zero and one half.

Core claim

Nachman's procedure is valid for reconstructing a conductivity γ ∈ H^{3/2,2n}(Ω) with γ ≡ 1 near ∂Ω from its Dirichlet-to-Neumann map Λ_γ, for n ≥ 3. A log-type stability estimate is obtained for γ ∈ H^{2-s,n/s}(Ω) when 0 < s < 1/2.

What carries the argument

Nachman's procedure, which recovers the conductivity from the Dirichlet-to-Neumann map.

If this is right

  • Unique reconstruction of the conductivity is possible from the Dirichlet-to-Neumann map under the given regularity and boundary condition.
  • A logarithmic stability estimate holds for conductivities in H^{2-s,n/s}(Ω) with 0 < s < 1/2.
  • The result applies in dimensions n at least 3 for smooth bounded domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This may permit the method to apply in physical settings where conductivity varies less smoothly than higher-regularity theorems require.
  • Similar reductions in required regularity could be investigated for other elliptic inverse problems that rely on boundary maps.

Load-bearing premise

The conductivity satisfies γ ∈ H^{3/2,2n}(Ω) and γ ≡ 1 near the boundary of a smooth bounded domain Ω ⊂ ℝ^n with n ≥ 3.

What would settle it

Two distinct conductivities both in H^{3/2,2n}(Ω) with γ ≡ 1 near the boundary that produce the same Dirichlet-to-Neumann map would show the procedure does not work.

read the original abstract

We show the validity of Nachman's procedure (Ann. Math. 128(3):531-576, 1988) for reconstructing a conductivity function $\gamma$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ($n\geq 3$) from its Dirichlet-to-Neumann map $\Lambda_\gamma$ for less regular conductivities, specifically $\gamma \in H^{3/2,2n}(\Omega)$ such that $\gamma \equiv 1$ near $\partial \Omega$. We also obtain a log-type stability estimate for the inverse problem when $\gamma$ has slightly higher regularity, i.e., $\gamma \in H^{2-s,n/s}(\Omega)$ for $0 < s <1/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends Nachman's 1988 reconstruction procedure to recover a conductivity γ ∈ H^{3/2,2n}(Ω) with γ ≡ 1 near ∂Ω from its Dirichlet-to-Neumann map Λ_γ, for n ≥ 3. It additionally derives a logarithmic stability estimate for the inverse problem when γ belongs to the slightly smoother class H^{2-s,n/s}(Ω) for 0 < s < 1/2.

Significance. If the central extension holds, the result is significant for the Calderón problem: it lowers the Sobolev regularity threshold at which Nachman's CGO-based reconstruction and scattering transform remain valid without modification, while the log-stability estimate quantifies the expected deterioration for rougher conductivities. The work directly addresses the gap between the classical C^2 or W^{2,p} settings and the minimal regularity needed for the DN map to be well-defined.

minor comments (3)
  1. The statement of the main theorem (presumably in §2 or §3) should explicitly record the precise Sobolev embedding or trace theorem used to justify that γ ≡ 1 near ∂Ω implies the boundary trace of the conductivity is 1, as this is used to normalize the CGO solutions.
  2. In the stability section, the dependence of the constant in the log estimate on s and n should be stated more explicitly; the current phrasing leaves open whether the constant blows up as s → 0.
  3. Notation for the space H^{3/2,2n}(Ω) is non-standard; a brief reminder of the definition (or reference to the precise mixed-norm Sobolev space) would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address point-by-point. The referee's summary accurately captures the main contributions.

Circularity Check

0 steps flagged

No circularity: extension of external 1988 procedure to new regularity class

full rationale

The paper's core claim is that Nachman's 1988 reconstruction procedure remains valid for conductivities in the Sobolev space H^{3/2,2n}(Ω) with γ ≡ 1 near ∂Ω. This is an independent verification that the requisite CGO solutions and scattering transform continue to be well-defined at this regularity threshold, rather than a reduction of the result to a fitted parameter, self-definition, or self-citation chain. The cited Nachman work is external (1988, non-overlapping authors), the log-stability estimate for the slightly smoother class is a standard consequence of the same framework, and no equations or steps in the provided abstract or description reduce the output to the input by construction. This is the expected non-finding for a pure-mathematics extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the stated Sobolev regularity of γ, the normalization γ ≡ 1 near the boundary, and the smoothness of the domain; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Ω is a smooth bounded domain in ℝ^n, n ≥ 3
    Required for the statement of both the reconstruction and the stability result.
  • domain assumption γ ≡ 1 near ∂Ω
    Explicitly required to make the reconstruction procedure valid at the stated regularity.

pith-pipeline@v0.9.0 · 5643 in / 1264 out tokens · 24659 ms · 2026-05-24T15:32:00.690781+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Stable determination of conductivity by bo undary measurements

    Giovanni Alessandrini. Stable determination of conductivity by bo undary measurements. Applicable Analysis, 27(1-3):153–172, 1988

    work page 1988

  2. [2]

    Singular solutions of elliptic equations and th e determination of conductivity by boundary measurements

    Giovanni Alessandrini. Singular solutions of elliptic equations and th e determination of conductivity by boundary measurements. Journal of Differential Equations , 84(2):252 – 272, 1990

    work page 1990

  3. [3]

    Determining Conductivity by Boundary Measurements, the St ability Issue , pages 317–324

    Giovanni Alessandrini. Determining Conductivity by Boundary Measurements, the St ability Issue , pages 317–324. Springer Netherlands, Dordrecht, 1991

    work page 1991

  4. [4]

    Reconstruction of the pote ntial from partial Cauchy data for the Schr¨ odinger equation.Indiana Univ

    Habib Ammari and Gunther Uhlmann. Reconstruction of the pote ntial from partial Cauchy data for the Schr¨ odinger equation.Indiana Univ. Math. J. , 53(1):169–183, 2004

    work page 2004

  5. [5]

    Assylbekov

    Yernat M. Assylbekov. Reconstruction in the partial data Calde r´ on problem on admissible manifolds. Inverse Probl. Imaging , 11(3):455–476, 2017

    work page 2017

  6. [6]

    Calder´ on’s inverse conductivity problem in the plane

    Kari Astala and Lassi P¨ aiv¨ arinta. Calder´ on’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1):265–299, 2006

    work page 2006

  7. [7]

    Imanuvilov, and M

    Eemeli Blaasten, Oleg Yu. Imanuvilov, and M. Yamamoto. Stability a nd uniqueness for a two- dimensional inverse boundary value problem for less regular potent ials. 2015

    work page 2015

  8. [8]

    R.M. Brown. Global uniqueness in the impedance-imaging problem fo r less regular conductivities. SIAM J. Math. Anal. , 27(4):1049–1056, July 1996

    work page 1996

  9. [9]

    Brown and Rodolfo H

    Russell M. Brown and Rodolfo H. Torres. Uniqueness in the invers e conductivity problem for conduc- tivities with 3 /2 derivatives in lp, p > 2n. Journal of Fourier Analysis and Applications , 9(6):563–574, Nov 2003

    work page 2003

  10. [10]

    Brown and Gunther A

    Russell M. Brown and Gunther A. Uhlmann. Uniqueness in the inve rse conductivity problem for non- smooth conductivities in two dimensions. Comm. Partial Differential Equations , 22(5-6):1009–1027, 1997

    work page 1997

  11. [11]

    A. L. Bukhgeim. Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl., 16(1):19–33, 2008

    work page 2008

  12. [12]

    Bukhgeim and Gunther Uhlmann

    Alexander L. Bukhgeim and Gunther Uhlmann. Recovering a pote ntial from partial Cauchy data. Comm. Partial Differential Equations , 27(3-4):653–668, 2002

    work page 2002

  13. [13]

    Calderon

    A.P. Calderon. On an inverse boundary value problem. Seminar on Numerical Analysis and its Appli- cations to Continuum Physics , pages 65–73, 1980. 22 ASHWIN TARIKERE

    work page 1980

  14. [14]

    Stab ility estimates for the Calder´ on problem with partial data

    Pedro Caro, David Dos Santos Ferreira, and Alberto Ruiz. Stab ility estimates for the Calder´ on problem with partial data. J. Differential Equations , 260(3):2457–2489, 2016

    work page 2016

  15. [15]

    Stability o f the caldern problem for less regular conductivities

    Pedro Caro, Andoni Garca, and Juan Manuel Reyes. Stability o f the caldern problem for less regular conductivities. Journal of Differential Equations , 254(2):469 – 492, 2013

    work page 2013

  16. [16]

    Inverse scatterin g for a random potential

    Pedro Caro, Tapio Helin, and Matti Lassas. Inverse scatterin g for a random potential. Analysis and Applications, 0(0):1–55, 0

    work page

  17. [17]

    Pedro Caro and Keith M. Rogers. Global uniqueness for the cald er´ on problem with lipschitz conductiv- ities. Forum of Mathematics, Pi , 4:e2, 2016

    work page 2016

  18. [18]

    Calder´ on prob lem: An introduction to inverse prob- lems

    Joel Feldman, Mikko Salo, and Gunther Uhlmann. Calder´ on prob lem: An introduction to inverse prob- lems

    work page

  19. [19]

    Reconstruction from boundar y measurements for less regular conduc- tivities

    Andoni Garc ´ ıa and Guo Zhang. Reconstruction from boundar y measurements for less regular conduc- tivities. Inverse Problems , 32(11):115015, oct 2016

    work page 2016

  20. [20]

    The Kato-Ponce inequality

    Loukas Grafakos and Seungly Oh. The Kato-Ponce inequality. Comm. Partial Differential Equations , 39(6):1128–1157, 2014

    work page 2014

  21. [21]

    Uniqueness in Calder´ on’s problem for conduct ivities with unbounded gradient

    Boaz Haberman. Uniqueness in Calder´ on’s problem for conduct ivities with unbounded gradient. Comm. Math. Phys. , 340(2):639–659, 2015

    work page 2015

  22. [22]

    Uniqueness in calderns prob lem with lipschitz conductivities

    Boaz Haberman and Daniel Tataru. Uniqueness in calderns prob lem with lipschitz conductivities. Duke Mathematical Journal , 162(3):497516, Feb 2013

    work page 2013

  23. [23]

    Stability estimates for the inverse conductivity pr oblem for less regular conductivities

    Horst Heck. Stability estimates for the inverse conductivity pr oblem for less regular conductivities. Communications in Partial Differential Equations , 34(2):107–118, 2009

    work page 2009

  24. [24]

    Stability estimates for the inve rse boundary value problem by partial Cauchy data

    Horst Heck and Jenn-Nan Wang. Stability estimates for the inve rse boundary value problem by partial Cauchy data. Inverse Problems , 22(5):1787–1796, 2006

    work page 2006

  25. [25]

    Imanuvilov and Masahiro Yamamoto

    Oleg Y. Imanuvilov and Masahiro Yamamoto. Uniqueness for inver se boundary value problems by Dirichlet-to-Neumann map on subboundaries. Milan J. Math. , 81(2):187–258, 2013

    work page 2013

  26. [26]

    Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto

    Oleg Yu. Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto. The Calder´ on problem with partial data in two dimensions. J. Amer. Math. Soc. , 23(3):655–691, 2010

    work page 2010

  27. [27]

    Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto

    Oleg Yu. Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto. Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets. Inverse Problems, 27(8):085007, 26, 2011

    work page 2011

  28. [28]

    On uniqueness in the inverse conductivity proble m with local data

    Victor Isakov. On uniqueness in the inverse conductivity proble m with local data. Inverse Probl. Imaging, 1(1):95–105, 2007

    work page 2007

  29. [29]

    Commutator estimates and the Euler and Navier-Stokes equations

    Tosio Kato and Gustavo Ponce. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. , 41(7):891–907, 1988

    work page 1988

  30. [30]

    The Calder´ on problem with partial da ta on manifolds and applications

    Carlos Kenig and Mikko Salo. The Calder´ on problem with partial da ta on manifolds and applications. Anal. PDE , 6(8):2003–2048, 2013

    work page 2003

  31. [31]

    Recent progress in the Calder´ on pr oblem with partial data

    Carlos Kenig and Mikko Salo. Recent progress in the Calder´ on pr oblem with partial data. In Inverse problems and applications , volume 615 of Contemp. Math. , pages 193–222. Amer. Math. Soc., Provi- dence, RI, 2014

    work page 2014

  32. [32]

    Kenig, Johannes Sj¨ ostrand, and Gunther Uhlmann

    Carlos E. Kenig, Johannes Sj¨ ostrand, and Gunther Uhlmann. The Calder´ on problem with partial data. Ann. of Math. (2) , 165(2):567–591, 2007

    work page 2007

  33. [33]

    The Calder´ on problem with partial data for less sm ooth conductivities

    Kim Knudsen. The Calder´ on problem with partial data for less sm ooth conductivities. Comm. Partial Differential Equations , 31(1-3):57–71, 2006

    work page 2006

  34. [34]

    The Calder´ on problem w ith partial data for conductivities with 3/2 derivatives

    Katya Krupchyk and Gunther Uhlmann. The Calder´ on problem w ith partial data for conductivities with 3/2 derivatives. Comm. Math. Phys. , 348(1):185–219, 2016

    work page 2016

  35. [35]

    Stability estimates for the inverse boundary value pr oblem by partial Cauchy data

    Ru-Yu Lai. Stability estimates for the inverse boundary value pr oblem by partial Cauchy data. Math. Methods Appl. Sci. , 38(8):1568–1581, 2015

    work page 2015

  36. [36]

    Nachman’s rec onstruction method for the calderon problem with discontinuous conductivities

    George Lytle, Peter Perry, and Samuli Siltanen. Nachman’s rec onstruction method for the calderon problem with discontinuous conductivities. Inverse Problems , 2019

    work page 2019

  37. [37]

    Exponential instability in an inverse problem fo r the schrdinger equation

    Niculae Mandache. Exponential instability in an inverse problem fo r the schrdinger equation. Inverse Problems, 17(5):1435–1444, aug 2001

    work page 2001

  38. [38]

    Reconstruction in the Calder ´ on problem with partial data

    Adrian Nachman and Brian Street. Reconstruction in the Calder ´ on problem with partial data. Comm. Partial Differential Equations , 35(2):375–390, 2010

    work page 2010

  39. [39]

    Adrian I. Nachman. Reconstructions from boundary measure ments. Annals of Mathematics , 128(3):531– 576, 1988. RECONSTRUCTION OF ROUGH CONDUCTIVITIES FROM BOUNDARY MEAS UREMENTS 23

    work page 1988

  40. [40]

    Adrian I. Nachman. Global uniqueness for a two-dimensional inv erse boundary value problem. Ann. of Math. (2) , 143(1):71–96, 1996

    work page 1996

  41. [41]

    Recovering a potential from cauchy data via complex geometrical optics solutions

    Hoai-Minh Nguyen and Daniel Spirn. Recovering a potential from cauchy data via complex geometrical optics solutions. 2014

    work page 2014

  42. [42]

    C omplex geometrical optics solutions for lipschitz conductivities

    Lassi Pivrinta, Alexander Panchenko, and Gunther Uhlmann. C omplex geometrical optics solutions for lipschitz conductivities. Rev. Mat. Iberoamericana, 19(1):57–72, 03 2003

    work page 2003

  43. [43]

    A partial data result for less regular conduc tivities in admissible geometries

    Casey Rodriguez. A partial data result for less regular conduc tivities in admissible geometries. Inverse Probl. Imaging, 10(1):247–262, 2016

    work page 2016

  44. [44]

    A global uniqueness theo rem for an inverse boundary value problem

    John Sylvester and Gunther Uhlmann. A global uniqueness theo rem for an inverse boundary value problem. Ann. of Math. (2) , 125(1):153–169, 1987

    work page 1987

  45. [45]

    H. Triebel. Theory of Function Spaces . Modern Birkh¨ auser Classics. Springer Basel, 2010

    work page 2010

  46. [46]

    Uniqueness in the Calder´ on problem with partial data for less smooth conductivities

    Guo Zhang. Uniqueness in the Calder´ on problem with partial data for less smooth conductivities. Inverse Problems, 28(10):105008, 18, 2012. Department of Mathematics, University of W ashington E-mail address : ashwin91@math.washington.edu

    work page 2012