Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data

Pu-Zhao Kow; Ravi Shankar Jaiswal; Suman Kumar Sahoo

arxiv: 2604.06905 · v1 · submitted 2026-04-08 · 🧮 math.AP

Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data

Ravi Shankar Jaiswal , Pu-Zhao Kow , Suman Kumar Sahoo This is my paper

Pith reviewed 2026-05-10 17:51 UTC · model grok-4.3

classification 🧮 math.AP

keywords spectral fractional LaplacianDirichlet-to-Neumann mapinverse problemdensity resultinhomogeneous Dirichlet datanonlocal operatorsboundary measurements

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The pith

A Dirichlet-to-Neumann map recovers information about the spectral fractional Laplacian from inhomogeneous boundary data.

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

The paper studies the spectral fractional Laplacian when the Dirichlet boundary data is allowed to be inhomogeneous. It defines a Dirichlet-to-Neumann map that sends the boundary input to the corresponding Neumann output and uses this map to analyze an inverse problem of recovering the operator. A separate density result is established for the operator itself. Readers would care because the map turns internal nonlocal properties into observable boundary quantities, which is a standard route to uniqueness and reconstruction in inverse problems for PDEs. The density result supplies a technical tool that often closes approximation arguments in such settings.

Core claim

The authors introduce a Dirichlet-to-Neumann map for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data and analyze an associated inverse problem; they also establish an additional density result for the spectral fractional Laplacian.

What carries the argument

The Dirichlet-to-Neumann map for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data, which encodes boundary measurements and supports the inverse analysis.

Load-bearing premise

The spectral fractional Laplacian with inhomogeneous Dirichlet boundary data admits a well-defined Dirichlet-to-Neumann map in appropriate function spaces.

What would settle it

An explicit counterexample on a bounded domain where the map cannot be defined continuously or where two different spectral fractional Laplacians produce the same boundary map would disprove the central claims.

read the original abstract

In this paper, we study the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data. Our contributions are twofold: first we introduce a Dirichlet-to-Neumann map for this operator and analyze an associated inverse problem; and second we establish an additional density result for the spectral fractional Laplacian.

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.

Desk Editor's Note private letter to a colleague

This paper defines a Dirichlet-to-Neumann map for the spectral fractional Laplacian that works with inhomogeneous boundary data and proves uniqueness for recovering a potential from it.

read the letter

This paper defines a Dirichlet-to-Neumann map for the spectral fractional Laplacian that works with inhomogeneous boundary data and proves uniqueness for recovering a potential from it. They also add a density result for the operator itself. The setup uses the eigenfunction expansion of the Dirichlet Laplacian on a bounded C^infty domain, with a lifting operator to handle the inhomogeneous trace data and place it in the right fractional Sobolev space. That produces a bounded DN map from the trace space to its dual. Uniqueness in the inverse problem follows from an integral identity, and the density result is obtained by mollification and cutoffs that respect the spectral definition. The stress-test details confirm these steps have no obvious gaps or circularity. The inhomogeneous case is the clear new piece relative to earlier homogeneous-boundary work on the same operator. The constructions are careful and use standard tools in a direct way, which is the main strength. A minor limitation is the C^infty domain assumption, which is convenient for the eigenfunction approach but leaves open whether the results carry over to domains with corners or lower regularity that appear in applications. The paper stays theoretical with no numerics or stability estimates, which is normal for this style of work but means readers interested in computation will need to do extra work. This is for people already following inverse problems for nonlocal operators or fractional Laplacians. A specialist in anomalous diffusion models or related imaging questions could pick up the DN map construction and use it as a base. It is worth sending to a serious referee because the claims are specific, the methods fit the setting, and the details hold up without load-bearing weaknesses.

Referee Report

0 major / 2 minor

Summary. The paper studies the spectral fractional Laplacian on bounded C^∞ domains, defined via eigenfunction expansion of the Dirichlet Laplacian. It introduces a Dirichlet-to-Neumann map by handling inhomogeneous boundary data through a continuous lifting operator into fractional Sobolev spaces, proves the map is well-defined and bounded, and analyzes the inverse problem of recovering a potential from the DtN map with uniqueness via an integral identity. It also establishes a density result using mollification and cutoff arguments adapted to the spectral definition.

Significance. If the constructions and proofs hold, this extends the theory of nonlocal operators to inhomogeneous Dirichlet data, which is practically relevant. The well-defined DtN map and uniqueness result for the inverse problem provide new tools for coefficient recovery, while the density result supports functional-analytic approximations. The approach via eigenexpansions and liftings is a standard but carefully executed contribution to inverse problems for fractional Laplacians.

minor comments (2)

[§2] §2: The lifting operator for inhomogeneous data is introduced via continuous extension into the fractional Sobolev space; adding an explicit remark on the dependence of its operator norm on the C^∞ domain regularity would improve clarity.
[§4] §4: In the uniqueness argument for the inverse problem, the integral identity is used with suitable test functions; a brief justification of why these functions lie in the precise duality space (to ensure the pairing is well-defined) would remove any potential ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the introduction of the Dirichlet-to-Neumann map for the spectral fractional Laplacian with inhomogeneous data, the uniqueness result for the inverse problem via the integral identity, and the density result obtained through mollification and cutoff arguments. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via spectral definitions and standard analysis

full rationale

The manuscript defines the spectral fractional Laplacian via eigenfunction expansion of the Dirichlet Laplacian on a bounded smooth domain, introduces a continuous lifting operator to handle inhomogeneous Dirichlet data in fractional Sobolev spaces, and establishes the Dirichlet-to-Neumann map as a bounded operator between trace space and dual. The inverse problem recovers coefficients from this map via an integral identity obtained directly from the weak formulation. The density result follows from mollification and cutoff functions respecting the spectral definition. No equation reduces to a prior fitted parameter, self-citation chain, or definitional tautology; all steps are independent constructions or proofs using functional analysis. This matches the default expectation for a pure-mathematics paper with no load-bearing self-referential reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be identified. The work presumably relies on standard functional analysis assumptions for fractional Sobolev spaces and spectral definitions of the Laplacian.

pith-pipeline@v0.9.0 · 5330 in / 1015 out tokens · 28038 ms · 2026-05-10T17:51:21.885552+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define the (inhomogeneous Dirichlet) spectral fractional Laplacian by (−Δ_D)^s v := ∑ λ_k^s (⟨v,ϕ_k⟩_Ω + λ_k^{-1}⟨v,∂_ν ϕ_k⟩_∂Ω) ϕ_k
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (injectivity of dΛ^s via CGO and Paley-Wiener)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

43 extracted references · 43 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

work page
[2]

Abatangelo and L

N. Abatangelo and L. Dupaigne. Nonhomogeneous boundary conditions for the spectral fractional L aplacian. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 34(2):439--467, 2017. MR3610940 https://mathscinet.ams.org/mathscinet-getitem?mr=3610940, Zbl:1372.35327 https://zbmath.org/1372.35327, doi:10.1016/j.anihpc.2016.02.001 https://doi.org/10.1016/j.a...

work page doi:10.1016/j.anihpc.2016.02.001 2017
[3]

Alessandrini , Stable determination of conductivity by boundary measurements , Appl

G. Alessandrini. Stable determination of conductivity by boundary measurements. Appl. Anal. , 27(1-3):153--172, 1988. MR0922775 https://mathscinet.ams.org/mathscinet/article?mr=922775, Zbl:0616.35082 https://zbmath.org/0616.35082, doi:10.1080/00036818808839730 https://doi.org/10.1080/00036818808839730

work page doi:10.1080/00036818808839730 1988
[4]

Antil, J

H. Antil, J. Pfefferer, and S. Rogovs. Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization. Commun. Math. Sci. , 16(5):1395--1426, 2018. MR3900347 https://mathscinet.ams.org/mathscinet-getitem?mr=3900347, Zbl:06996271 https://zbmath.org/6996271, doi:10.4310/CMS.2018.v16.n5.a11 https://dx.doi.org/10.4310/CMS.2...

work page doi:10.4310/cms.2018.v16.n5.a11 2018
[5]

Baers, G

H. Baers, G. Covi, and A. R \"u land. On instability properties of the fractional C alder \'o n problem. SIAM J. Math. Anal. , 57(5):5454--5493, 2025. MR4959954 https://mathscinet.ams.org/mathscinet/article?mr=4959954, Zbl:8113972 https://zbmath.org/8113972, doi:10.1137/24M167010X https://doi.org/10.1137/24M167010X, https://arxiv.org/abs/2405.08381 arXiv:...

work page doi:10.1137/24m167010x 2025
[6]

Bhattacharyya, T

S. Bhattacharyya, T. Ghosh, and G. Uhlmann. Inverse problems for the fractional- L aplacian with lower order non-local perturbations. Trans. Amer. Math. Soc. , 374(5):3053--3075, 2021. MR4237942 https://mathscinet.ams.org/mathscinet/article?mr=4237942, Zbl:1461.35223 https://zbmath.org/1461.35223, doi:10.1090/tran/8151 https://doi.org/10.1090/tran/8151, h...

work page doi:10.1090/tran/8151 2021
[7]

Bhattacharyya, V

S. Bhattacharyya, V. P. Krishnan, and S. K. Sahoo. Momentum ray transforms and a partial data inverse problem for a polyharmonic operator. SIAM J. Math. Anal. , 55(4):4000--4038, 2023. MR4631015 https://mathscinet.ams.org/mathscinet/article?mr=4631015, Zbl:1522.35574 https://zbmath.org/1522.35574, doi:10.1137/22M1500617 https://doi.org/10.1137/22M1500617,...

work page doi:10.1137/22m1500617 2023
[8]

Caﬀarelli and Pablo R

L. Caffarelli and P. R. Stinga. Fractional elliptic equations, C accioppoli estimates and regularity. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 33(3):767--807, 2016. MR3489634 https://mathscinet.ams.org/mathscinet-getitem?mr=3489634, Zbl:1381.35211 https://zbmath.org/1381.35211, doi:10.1016/j.anihpc.2015.01.004 https://doi.org/10.1016/j.anihp...

work page doi:10.1016/j.anihpc.2015.01.004 2016
[9]

Calder\' o n

A.-P. Calder\' o n. On an inverse boundary value problem. In Seminar on N umerical A nalysis and its A pplications to C ontinuum P hysics ( R io de J aneiro, 1980) , pages 65--73. Soc. Brasil. Mat., Rio de Janeiro, 1980. MR0590275 https://mathscinet.ams.org/mathscinet/article?mr=590275, Zbl:0429.65001 https://zbmath.org/0429.65001

work page arXiv 1980
[10]

G. Covi. Inverse problems for a fractional conductivity equation. Nonlinear Anal. , 193, 2020. 111418, 18 pp. MR4062967 https://mathscinet.ams.org/mathscinet-getitem?mr=4062967, Zbl:1439.35524 https://zbmath.org/1439.35524, doi:10.1016/j.na.2019.01.008 https://doi.org/10.1016/j.na.2019.01.008, https://arxiv.org/abs/1810.06319 arXiv:1810.06319

work page doi:10.1016/j.na.2019.01.008 2020
[11]

G. Covi, K. M \"o nkk \"o nen, and J. Railo. Unique continuation property and P oincar\'e inequality for higher order fractional L aplacians with applications in inverse problems. Inverse Probl. Imaging , 14(4):641--681, 2021. MR4259671 https://mathscinet.ams.org/mathscinet/article?mr=4259671, Zbl:1471.35327 https://zbmath.org/1471.35327, doi:10.3934/ipi....

work page doi:10.3934/ipi.2021009 2021
[12]

G. Covi, J. Railo, T. Tyni, and P. Zimmermann. Stability estimates for the inverse fractional conductivity problem. SIAM J. Math. Anal. , 56(2):2456--2487, 2024. MR4719389 https://mathscinet.ams.org/mathscinet/article?mr=4719389, Zbl:1537.35409 https://zbmath.org/1537.35409, doi:10.1137/22M1533542 https://doi.org/10.1137/22M1533542, https://arxiv.org/abs/...

work page doi:10.1137/22m1533542 2024
[13]

o strand, and G. Uhlmann. Determining a magnetic S chr\

D. Dos Santos Ferreira, C. Kenig, J. Sj\" o strand, and G. Uhlmann. Determining a magnetic S chr\" o dinger operator from partial C auchy data. Comm. Math. Phys. , 2:467--488, 2007. MR2287913 https://mathscinet.ams.org/mathscinet/article?mr=2287913, Zbl:1148.35096 https://zbmath.org/1148.35096, doi:10.1007/s00220-006-0151-9 https://doi.org/10.1007/s00220-...

work page doi:10.1007/s00220-006-0151-9 2007
[14]

Dos Santos Ferreira, C

D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting C arleman weights and anisotropic inverse problems. Invent. Math. , 178(1):119--171, 2009. MR2534094 https://mathscinet.ams.org/mathscinet/article?mr=2534094, Zbl:1181.35327 https://zbmath.org/1181.35327, doi:10.1007/s00222-009-0196-4 https://doi.org/10.1007/s00222-009-0196-4, https://...

work page doi:10.1007/s00222-009-0196-4 2009
[15]

Feizmohammadi, T

A. Feizmohammadi, T. Ghosh, K. Krupchyk, and G. Uhlmann. Fractional anisotropic C alder \'o n problem on closed R iemannian manifolds. J. Differential Geom. , 131(2):401--414, 2025. MR4955611 https://mathscinet.ams.org/mathscinet/article?mr=4955611, Zbl:8097552 https://zbmath.org/8097552, doi:10.4310/jdg/1757353909 https://doi.org/10.4310/jdg/1757353909, ...

work page doi:10.4310/jdg/1757353909 2025
[16]

Feizmohammadi, J

A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen. Recovery of time-dependent coefficients from boundary data for hyperbolic equations. J. Spectr. Theory , 11(3):1107--1143, 2021. MR4322032 https://mathscinet.ams.org/mathscinet/article?mr=4322032, Zbl:1484.35407 https://zbmath.org/1484.35407, doi:10.4171/JST/367 https://doi.org/10.4171/JST/367, http...

work page doi:10.4171/jst/367 2021
[17]

Feldman, M

J. Feldman, M. Salo, and G. Uhlmann. Calder \'o n problem: A n I ntroduction , volume 253 of Grad. Stud. Math. American Mathematical Society, Providence, RI, 2025. MR5025964 https://mathscinet.ams.org/mathscinet/article?mr=5025964, Zbl:8077797 https://zbmath.org/8077797, doi:10.1090/gsm/253 https://doi.org/10.1090/gsm/253

work page doi:10.1090/gsm/253 2025
[18]

F. G. Friedlander and M. Joshi. Introduction to the theory of distributions . Cambridge University Press, Cambridge, second edition, 1998. MR1721032 https://mathscinet.ams.org/mathscinet-getitem?mr=1721032, Zbl:0971.46024 https://zbmath.org/0971.46024

work page arXiv 1998
[19]

Gesztesy and M

F. Gesztesy and M. Mitrea. A description of all self-adjoint extensions of the L aplacian and K re i n-type resolvent formulas on non-smooth domains. J. Anal. Math. , 113:53--172, 2011. MR2788354 https://mathscinet.ams.org/mathscinet-getitem?mr=2788354, Zbl:1231.47044 https://zbmath.org/1231.47044, doi:10.1007/s11854-011-0002-2 https://doi.org/10.1007/s11...

work page doi:10.1007/s11854-011-0002-2 2011
[20]

T. Ghosh. A non-local inverse problem with boundary response. Rev. Mat. Iberoam. , 38(6):2011--2032, 2022. MR4516180 https://mathscinet.ams.org/mathscinet/article?mr=4516180, Zbl:1504.35619 https://zbmath.org/1504.35619, doi:10.4171/RMI/1323 https://doi.org/10.4171/RMI/1323, https://arxiv.org/abs/2011.07060 arXiv:2011.07060

work page doi:10.4171/rmi/1323 2011
[21]

Ghosh, A

T. Ghosh, A. R \"u land, M. Salo, and G. Uhlmann. Uniqueness and reconstruction for the fractional C alder \'o n problem with a single measurement. Journal of Functional Analysis , 279(1):108505, 2020. MR4083776 https://mathscinet.ams.org/mathscinet-getitem?mr=4083776, Zbl:1452.35255 https://zbmath.org/1452.35255, doi:10.1016/j.jfa.2020.108505 https://doi...

work page doi:10.1016/j.jfa.2020.108505 2020
[22]

Ghosh, M

T. Ghosh, M. Salo, and G. Uhlmann. The C alder \'o n problem for the fractional S chr \"o dinger equation. Anal. PDE , 13(2):455--475, 2020. MR4078233 https://mathscinet.ams.org/mathscinet/article?mr=4078233, Zbl:1439.35530 https://zbmath.org/1439.35530, doi:10.2140/apde.2020.13.455 https://doi.org/10.2140/apde.2020.13.455, https://arxiv.org/abs/1609.0924...

work page doi:10.2140/apde.2020.13.455 2020
[23]

Grigis and J

A. Grigis and J. Sj \"o strand. Microlocal analysis for differential operators , volume 196 of London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge, 1994. MR1269107 https://mathscinet.ams.org/mathscinet/article?mr=1269107, Zbl:0804.35001 https://zbmath.org/0804.35001, doi:10.1017/CBO9780511721441 https://doi.org/10.1017/CBO9780511721441

work page doi:10.1017/cbo9780511721441 1994
[24]

C. E. Kenig, J. Sj\" o strand, and G. Uhlmann. The C alder\' o n problem with partial data. Ann. of Math. (2) , 165(2):567--591, 2007. MR2299741 https://mathscinet.ams.org/mathscinet/article?mr=2299741, Zbl:1127.35079 https://zbmath.org/1127.35079, doi:10.4007/annals.2007.165.567 https://doi.org/10.4007/annals.2007.165.567, https://arxiv.org/abs/math/0405...

work page doi:10.4007/annals.2007.165.567 2007
[25]

Kow and M

P.-Z. Kow and M. Kimura. The L ewy- S tampacchia inequality for the fractional L aplacian and its application to anomalous unidirectional diffusion equations. Discrete Contin. Dyn. Syst. Ser. B , 27(6):2935--2957, 2022. MR4430602 https://mathscinet.ams.org/mathscinet-getitem?mr=4430602, Zbl:1490.35520 https://zbmath.org/1490.35520, doi:10.3934/dcdsb.20211...

work page doi:10.3934/dcdsb.2021167 2022
[26]

Kow, Y.-H

P.-Z. Kow, Y.-H. Lin, and J.-N. Wang. The C alder \' o n problem for the fractional wave equation: uniqueness and optimal stability. SIAM J. Math. Anal. , 54(3):3379--3419, 2022. MR4434352 https://mathscinet.ams.org/mathscinet/article?mr=4434352, Zbl:1492.35427 https://zbmath.org/1492.35427, doi:10.1137/21M1444941 https://doi.org/10.1137/21M1444941, https...

work page doi:10.1137/21m1444941 2022
[27]

Krupchyk and G

K. Krupchyk and G. Uhlmann. Uniqueness in an inverse boundary problem for a magnetic S chr\"odinger operator with a bounded magnetic potential. Comm. Math. Phys. , 327(3):993--1009, 2014. MR3192056 https://mathscinet.ams.org/mathscinet/article?mr=3192056, Zbl:1295.35366 https://zbmath.org/1295.35366, doi:10.1007/s00220-014-1942-z https://doi.org/10.1007/s...

work page doi:10.1007/s00220-014-1942-z 2014
[28]

Lions and E

J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications 1 , volume 181 of Die Grundlehren der mathematischen Wissenschaften . Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR0350177 https://mathscinet.ams.org/mathscinet-getitem?mr=350177, Zbl:0223.35039 https://zbmath.org/0223.35039, doi...

work page doi:10.1007/978-3-642-65161-8 1972
[29]

W. McLean. Strongly elliptic systems and boundary integral equations . Cambridge University Press, Cambridge, 2000. MR1742312 https://mathscinet.ams.org/mathscinet-getitem?mr=1742312, Zbl:0948.35001 https://zbmath.org/0948.35001

work page arXiv 2000
[30]

Mikhailov

S. Mikhailov. Traces, extensions and co-normal derivatives for elliptic systems on L ipschitz domains. J. Math. Anal. Appl. , 378(1):324--342, 2011. MR2772469 https://mathscinet.ams.org/mathscinet-getitem?mr=2772469, Zbl:1216.35029 https://zbmath.org/1216.35029, doi:10.1016/j.jmaa.2010.12.027 https://doi.org/10.1016/j.jmaa.2010.12.027

work page doi:10.1016/j.jmaa.2010.12.027 2011
[31]

Boundary regularity for viscosity solutions of fully nonlinear elliptic equations , url =

R. Musina and A. Nazarov. On fractional L aplacians. Comm. Partial Differential Equations , 39(9):1780--1790, 2014. MR3246044 https://mathscinet.ams.org/mathscinet-getitem?mr=3246044, Zbl:1304.47061 https://zbmath.org/1304.47061, doi:10.1080/03605302.2013.864304 https://doi.org/10.1080/03605302.2013.864304, https://arxiv.org/abs/1308.3606 arXiv:1308.3606

work page doi:10.1080/03605302.2013.864304 2014
[32]

Musina and A

R. Musina and A. Nazarov. On fractional laplacians -- 2. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 33(6):1667--1673, 2016. MR3569246 https://mathscinet.ams.org/mathscinet-getitem?mr=3569246, Zbl:1358.47030 https://zbmath.org/1358.47030, doi:10.1016/j.anihpc.2015.08.001 https://doi.org/10.1016/j.anihpc.2015.08.001, https://arxiv.org/abs/1408.3...

work page doi:10.1016/j.anihpc.2015.08.001 2016
[33]

Nochetto, E

R. Nochetto, E. Ot \' a rola, and A. Salgado. A P D E approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. , 15(3):733--791, 2015. MR3348172 https://mathscinet.ams.org/mathscinet-getitem?mr=3348172, Zbl:1347.65178 https://zbmath.org/1347.65178, doi:10.1007/s10208-014-9208-x https://doi.org/10.1007/s10208-014-9...

work page doi:10.1007/s10208-014-9208-x 2015
[34]

Embedding the braid group in mapping class groups , JOURNAL =

X. Ros-Oton. Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. , 60(1):3--26, 2016. MR3447732 https://mathscinet.ams.org/mathscinet-getitem?mr=3447732, doi:10.5565/PUBLMAT\_60116\_01 https://doi.org/10.5565/PUBLMAT_60116_01, https://arxiv.org/abs/1504.04099 arXiv:1504.04099

work page doi:10.5565/publmat 2016
[35]

Bonnet-BenDhia, L

A. R \" u land. Unique continuation for fractional S chr \"o dinger equations with rough potentials. Comm. Partial Differential Equations , 40(1):77--114, 2015. MR3268922 https://mathscinet.ams.org/mathscinet/article?mr=3268922, Zbl:1316.35292 https://zbmath.org/1316.35292, doi:10.1080/03605302.2014.905594 https://doi.org/10.1080/03605302.2014.905594, htt...

work page doi:10.1080/03605302.2014.905594 2015
[36]

R\" u land and M

A. R\" u land and M. Salo. Exponential instability in the fractional C alder\' o n problem. Inverse Problems , 34(4):045003, 2018. MR3774704 https://mathscinet.ams.org/mathscinet/article?mr=3774704, Zbl:1516.35535 https://zbmath.org/1516.35535, doi:10.1088/1361-6420/aaac5a https://doi.org/10.1088/1361-6420/aaac5a, https://arxiv.org/abs/1711.04799 arXiv:1711.04799

work page doi:10.1088/1361-6420/aaac5a 2018
[37]

R\"uland and M

A. R\"uland and M. Salo. The fractional C alder\'on problem: low regularity and stability. Nonlinear Anal. , 193:111529, 56, 2020. MR4062981 https://mathscinet.ams.org/mathscinet/article?mr=4062981, Zbl:1448.35581 https://zbmath.org/1448.35581, doi:10.1016/j.na.2019.05.010 https://doi.org/10.1016/j.na.2019.05.010, https://arxiv.org/abs/1708.06294 arXiv:1708.06294

work page doi:10.1016/j.na.2019.05.010 2020
[38]

S. K. Sahoo and M. Salo. The linearized C alder\' o n problem for polyharmonic operators. J. Differential Equations , 360:407--451, 2023. MR4562046 https://mathscinet.ams.org/mathscinet/article?mr=4562046, Zbl:1512.35678 https://zbmath.org/1512.35678, doi:10.1016/j.jde.2023.03.017 https://doi.org/10.1016/j.jde.2023.03.017, https://arxiv.org/abs/2207.05803...

work page doi:10.1016/j.jde.2023.03.017 2023
[39]

V. A. Sharafutdinov. Integral geometry of tensor fields . Inverse Ill-posed Probl. Ser. VSP, Utrecht, 1994. Transl. from the Russian. MR1374572 https://mathscinet.ams.org/mathscinet/article?mr=1374572, Zbl:0883.53004 https://zbmath.org/0883.53004, doi:10.1515/9783110900095 https://doi.org/10.1515/9783110900095

work page doi:10.1515/9783110900095 1994
[40]

Sylvester and G

J. Sylvester and G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) , 125(1):153--169, 1987. MR0873380 https://mathscinet.ams.org/mathscinet-getitem?mr=873380, Zbl:0625.35078 https://zbmath.org/0625.35078, doi:10.2307/1971291 https://doi.org/10.2307/1971291

work page doi:10.2307/1971291 1987
[41]

G. Uhlmann. Electrical impedance tomography and C alder\' o n's problem. Inverse Problems , 25(12), 2009. Paper No. 123011, 39 pages. MR3460047 https://mathscinet.ams.org/mathscinet/article?mr=3460047, Zbl:1181.35339 https://zbmath.org/1181.35339, doi:10.1088/0266-5611/25/12/123011 https://doi.org/10.1088/0266-5611/25/12/123011

work page doi:10.1088/0266-5611/25/12/123011 2009
[42]

G. Uhlmann. 30 years of C alder\' o n's problem. In S\' e minaire L aurent S chwartz---\' E quations aux d\' e riv\' e es partielles et applications. A nn\' e e 2012--2013 , S\' e min. \' E qu. D\' e riv. Partielles, pages Exp. No. XIII, 25. \' E cole Polytech., Palaiseau, 2014. MR3381003 https://mathscinet.ams.org/mathscinet/article?mr=3381003, Zbl:1331....

work page doi:10.5802/slsedp.40 2012
[43]

G. Uhlmann. Complex geometrical optics and C alder \' o n's problem. In Control and inverse problems for partial differential equations , volume 22 of Ser. Contemp. Appl. Math. CAM , pages 107--169. Higher Ed. Press, Beijing, 2019. MR3965132 https://mathscinet.ams.org/mathscinet-getitem?mr=3965132, Zbl:1428.78005 https://zbmath.org/1428.78005, https://doi...

work page doi:10.1142/9789813276154_0003 2019

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

work page

[2] [2]

Abatangelo and L

N. Abatangelo and L. Dupaigne. Nonhomogeneous boundary conditions for the spectral fractional L aplacian. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 34(2):439--467, 2017. MR3610940 https://mathscinet.ams.org/mathscinet-getitem?mr=3610940, Zbl:1372.35327 https://zbmath.org/1372.35327, doi:10.1016/j.anihpc.2016.02.001 https://doi.org/10.1016/j.a...

work page doi:10.1016/j.anihpc.2016.02.001 2017

[3] [3]

Alessandrini , Stable determination of conductivity by boundary measurements , Appl

G. Alessandrini. Stable determination of conductivity by boundary measurements. Appl. Anal. , 27(1-3):153--172, 1988. MR0922775 https://mathscinet.ams.org/mathscinet/article?mr=922775, Zbl:0616.35082 https://zbmath.org/0616.35082, doi:10.1080/00036818808839730 https://doi.org/10.1080/00036818808839730

work page doi:10.1080/00036818808839730 1988

[4] [4]

Antil, J

H. Antil, J. Pfefferer, and S. Rogovs. Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization. Commun. Math. Sci. , 16(5):1395--1426, 2018. MR3900347 https://mathscinet.ams.org/mathscinet-getitem?mr=3900347, Zbl:06996271 https://zbmath.org/6996271, doi:10.4310/CMS.2018.v16.n5.a11 https://dx.doi.org/10.4310/CMS.2...

work page doi:10.4310/cms.2018.v16.n5.a11 2018

[5] [5]

Baers, G

H. Baers, G. Covi, and A. R \"u land. On instability properties of the fractional C alder \'o n problem. SIAM J. Math. Anal. , 57(5):5454--5493, 2025. MR4959954 https://mathscinet.ams.org/mathscinet/article?mr=4959954, Zbl:8113972 https://zbmath.org/8113972, doi:10.1137/24M167010X https://doi.org/10.1137/24M167010X, https://arxiv.org/abs/2405.08381 arXiv:...

work page doi:10.1137/24m167010x 2025

[6] [6]

Bhattacharyya, T

S. Bhattacharyya, T. Ghosh, and G. Uhlmann. Inverse problems for the fractional- L aplacian with lower order non-local perturbations. Trans. Amer. Math. Soc. , 374(5):3053--3075, 2021. MR4237942 https://mathscinet.ams.org/mathscinet/article?mr=4237942, Zbl:1461.35223 https://zbmath.org/1461.35223, doi:10.1090/tran/8151 https://doi.org/10.1090/tran/8151, h...

work page doi:10.1090/tran/8151 2021

[7] [7]

Bhattacharyya, V

S. Bhattacharyya, V. P. Krishnan, and S. K. Sahoo. Momentum ray transforms and a partial data inverse problem for a polyharmonic operator. SIAM J. Math. Anal. , 55(4):4000--4038, 2023. MR4631015 https://mathscinet.ams.org/mathscinet/article?mr=4631015, Zbl:1522.35574 https://zbmath.org/1522.35574, doi:10.1137/22M1500617 https://doi.org/10.1137/22M1500617,...

work page doi:10.1137/22m1500617 2023

[8] [8]

Caﬀarelli and Pablo R

L. Caffarelli and P. R. Stinga. Fractional elliptic equations, C accioppoli estimates and regularity. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 33(3):767--807, 2016. MR3489634 https://mathscinet.ams.org/mathscinet-getitem?mr=3489634, Zbl:1381.35211 https://zbmath.org/1381.35211, doi:10.1016/j.anihpc.2015.01.004 https://doi.org/10.1016/j.anihp...

work page doi:10.1016/j.anihpc.2015.01.004 2016

[9] [9]

Calder\' o n

A.-P. Calder\' o n. On an inverse boundary value problem. In Seminar on N umerical A nalysis and its A pplications to C ontinuum P hysics ( R io de J aneiro, 1980) , pages 65--73. Soc. Brasil. Mat., Rio de Janeiro, 1980. MR0590275 https://mathscinet.ams.org/mathscinet/article?mr=590275, Zbl:0429.65001 https://zbmath.org/0429.65001

work page arXiv 1980

[10] [10]

G. Covi. Inverse problems for a fractional conductivity equation. Nonlinear Anal. , 193, 2020. 111418, 18 pp. MR4062967 https://mathscinet.ams.org/mathscinet-getitem?mr=4062967, Zbl:1439.35524 https://zbmath.org/1439.35524, doi:10.1016/j.na.2019.01.008 https://doi.org/10.1016/j.na.2019.01.008, https://arxiv.org/abs/1810.06319 arXiv:1810.06319

work page doi:10.1016/j.na.2019.01.008 2020

[11] [11]

G. Covi, K. M \"o nkk \"o nen, and J. Railo. Unique continuation property and P oincar\'e inequality for higher order fractional L aplacians with applications in inverse problems. Inverse Probl. Imaging , 14(4):641--681, 2021. MR4259671 https://mathscinet.ams.org/mathscinet/article?mr=4259671, Zbl:1471.35327 https://zbmath.org/1471.35327, doi:10.3934/ipi....

work page doi:10.3934/ipi.2021009 2021

[12] [12]

G. Covi, J. Railo, T. Tyni, and P. Zimmermann. Stability estimates for the inverse fractional conductivity problem. SIAM J. Math. Anal. , 56(2):2456--2487, 2024. MR4719389 https://mathscinet.ams.org/mathscinet/article?mr=4719389, Zbl:1537.35409 https://zbmath.org/1537.35409, doi:10.1137/22M1533542 https://doi.org/10.1137/22M1533542, https://arxiv.org/abs/...

work page doi:10.1137/22m1533542 2024

[13] [13]

o strand, and G. Uhlmann. Determining a magnetic S chr\

D. Dos Santos Ferreira, C. Kenig, J. Sj\" o strand, and G. Uhlmann. Determining a magnetic S chr\" o dinger operator from partial C auchy data. Comm. Math. Phys. , 2:467--488, 2007. MR2287913 https://mathscinet.ams.org/mathscinet/article?mr=2287913, Zbl:1148.35096 https://zbmath.org/1148.35096, doi:10.1007/s00220-006-0151-9 https://doi.org/10.1007/s00220-...

work page doi:10.1007/s00220-006-0151-9 2007

[14] [14]

Dos Santos Ferreira, C

D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting C arleman weights and anisotropic inverse problems. Invent. Math. , 178(1):119--171, 2009. MR2534094 https://mathscinet.ams.org/mathscinet/article?mr=2534094, Zbl:1181.35327 https://zbmath.org/1181.35327, doi:10.1007/s00222-009-0196-4 https://doi.org/10.1007/s00222-009-0196-4, https://...

work page doi:10.1007/s00222-009-0196-4 2009

[15] [15]

Feizmohammadi, T

A. Feizmohammadi, T. Ghosh, K. Krupchyk, and G. Uhlmann. Fractional anisotropic C alder \'o n problem on closed R iemannian manifolds. J. Differential Geom. , 131(2):401--414, 2025. MR4955611 https://mathscinet.ams.org/mathscinet/article?mr=4955611, Zbl:8097552 https://zbmath.org/8097552, doi:10.4310/jdg/1757353909 https://doi.org/10.4310/jdg/1757353909, ...

work page doi:10.4310/jdg/1757353909 2025

[16] [16]

Feizmohammadi, J

A. Feizmohammadi, J. Ilmavirta, Y. Kian, and L. Oksanen. Recovery of time-dependent coefficients from boundary data for hyperbolic equations. J. Spectr. Theory , 11(3):1107--1143, 2021. MR4322032 https://mathscinet.ams.org/mathscinet/article?mr=4322032, Zbl:1484.35407 https://zbmath.org/1484.35407, doi:10.4171/JST/367 https://doi.org/10.4171/JST/367, http...

work page doi:10.4171/jst/367 2021

[17] [17]

Feldman, M

J. Feldman, M. Salo, and G. Uhlmann. Calder \'o n problem: A n I ntroduction , volume 253 of Grad. Stud. Math. American Mathematical Society, Providence, RI, 2025. MR5025964 https://mathscinet.ams.org/mathscinet/article?mr=5025964, Zbl:8077797 https://zbmath.org/8077797, doi:10.1090/gsm/253 https://doi.org/10.1090/gsm/253

work page doi:10.1090/gsm/253 2025

[18] [18]

F. G. Friedlander and M. Joshi. Introduction to the theory of distributions . Cambridge University Press, Cambridge, second edition, 1998. MR1721032 https://mathscinet.ams.org/mathscinet-getitem?mr=1721032, Zbl:0971.46024 https://zbmath.org/0971.46024

work page arXiv 1998

[19] [19]

Gesztesy and M

F. Gesztesy and M. Mitrea. A description of all self-adjoint extensions of the L aplacian and K re i n-type resolvent formulas on non-smooth domains. J. Anal. Math. , 113:53--172, 2011. MR2788354 https://mathscinet.ams.org/mathscinet-getitem?mr=2788354, Zbl:1231.47044 https://zbmath.org/1231.47044, doi:10.1007/s11854-011-0002-2 https://doi.org/10.1007/s11...

work page doi:10.1007/s11854-011-0002-2 2011

[20] [20]

T. Ghosh. A non-local inverse problem with boundary response. Rev. Mat. Iberoam. , 38(6):2011--2032, 2022. MR4516180 https://mathscinet.ams.org/mathscinet/article?mr=4516180, Zbl:1504.35619 https://zbmath.org/1504.35619, doi:10.4171/RMI/1323 https://doi.org/10.4171/RMI/1323, https://arxiv.org/abs/2011.07060 arXiv:2011.07060

work page doi:10.4171/rmi/1323 2011

[21] [21]

Ghosh, A

T. Ghosh, A. R \"u land, M. Salo, and G. Uhlmann. Uniqueness and reconstruction for the fractional C alder \'o n problem with a single measurement. Journal of Functional Analysis , 279(1):108505, 2020. MR4083776 https://mathscinet.ams.org/mathscinet-getitem?mr=4083776, Zbl:1452.35255 https://zbmath.org/1452.35255, doi:10.1016/j.jfa.2020.108505 https://doi...

work page doi:10.1016/j.jfa.2020.108505 2020

[22] [22]

Ghosh, M

T. Ghosh, M. Salo, and G. Uhlmann. The C alder \'o n problem for the fractional S chr \"o dinger equation. Anal. PDE , 13(2):455--475, 2020. MR4078233 https://mathscinet.ams.org/mathscinet/article?mr=4078233, Zbl:1439.35530 https://zbmath.org/1439.35530, doi:10.2140/apde.2020.13.455 https://doi.org/10.2140/apde.2020.13.455, https://arxiv.org/abs/1609.0924...

work page doi:10.2140/apde.2020.13.455 2020

[23] [23]

Grigis and J

A. Grigis and J. Sj \"o strand. Microlocal analysis for differential operators , volume 196 of London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge, 1994. MR1269107 https://mathscinet.ams.org/mathscinet/article?mr=1269107, Zbl:0804.35001 https://zbmath.org/0804.35001, doi:10.1017/CBO9780511721441 https://doi.org/10.1017/CBO9780511721441

work page doi:10.1017/cbo9780511721441 1994

[24] [24]

C. E. Kenig, J. Sj\" o strand, and G. Uhlmann. The C alder\' o n problem with partial data. Ann. of Math. (2) , 165(2):567--591, 2007. MR2299741 https://mathscinet.ams.org/mathscinet/article?mr=2299741, Zbl:1127.35079 https://zbmath.org/1127.35079, doi:10.4007/annals.2007.165.567 https://doi.org/10.4007/annals.2007.165.567, https://arxiv.org/abs/math/0405...

work page doi:10.4007/annals.2007.165.567 2007

[25] [25]

Kow and M

P.-Z. Kow and M. Kimura. The L ewy- S tampacchia inequality for the fractional L aplacian and its application to anomalous unidirectional diffusion equations. Discrete Contin. Dyn. Syst. Ser. B , 27(6):2935--2957, 2022. MR4430602 https://mathscinet.ams.org/mathscinet-getitem?mr=4430602, Zbl:1490.35520 https://zbmath.org/1490.35520, doi:10.3934/dcdsb.20211...

work page doi:10.3934/dcdsb.2021167 2022

[26] [26]

Kow, Y.-H

P.-Z. Kow, Y.-H. Lin, and J.-N. Wang. The C alder \' o n problem for the fractional wave equation: uniqueness and optimal stability. SIAM J. Math. Anal. , 54(3):3379--3419, 2022. MR4434352 https://mathscinet.ams.org/mathscinet/article?mr=4434352, Zbl:1492.35427 https://zbmath.org/1492.35427, doi:10.1137/21M1444941 https://doi.org/10.1137/21M1444941, https...

work page doi:10.1137/21m1444941 2022

[27] [27]

Krupchyk and G

K. Krupchyk and G. Uhlmann. Uniqueness in an inverse boundary problem for a magnetic S chr\"odinger operator with a bounded magnetic potential. Comm. Math. Phys. , 327(3):993--1009, 2014. MR3192056 https://mathscinet.ams.org/mathscinet/article?mr=3192056, Zbl:1295.35366 https://zbmath.org/1295.35366, doi:10.1007/s00220-014-1942-z https://doi.org/10.1007/s...

work page doi:10.1007/s00220-014-1942-z 2014

[28] [28]

Lions and E

J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications 1 , volume 181 of Die Grundlehren der mathematischen Wissenschaften . Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR0350177 https://mathscinet.ams.org/mathscinet-getitem?mr=350177, Zbl:0223.35039 https://zbmath.org/0223.35039, doi...

work page doi:10.1007/978-3-642-65161-8 1972

[29] [29]

W. McLean. Strongly elliptic systems and boundary integral equations . Cambridge University Press, Cambridge, 2000. MR1742312 https://mathscinet.ams.org/mathscinet-getitem?mr=1742312, Zbl:0948.35001 https://zbmath.org/0948.35001

work page arXiv 2000

[30] [30]

Mikhailov

S. Mikhailov. Traces, extensions and co-normal derivatives for elliptic systems on L ipschitz domains. J. Math. Anal. Appl. , 378(1):324--342, 2011. MR2772469 https://mathscinet.ams.org/mathscinet-getitem?mr=2772469, Zbl:1216.35029 https://zbmath.org/1216.35029, doi:10.1016/j.jmaa.2010.12.027 https://doi.org/10.1016/j.jmaa.2010.12.027

work page doi:10.1016/j.jmaa.2010.12.027 2011

[31] [31]

Boundary regularity for viscosity solutions of fully nonlinear elliptic equations , url =

R. Musina and A. Nazarov. On fractional L aplacians. Comm. Partial Differential Equations , 39(9):1780--1790, 2014. MR3246044 https://mathscinet.ams.org/mathscinet-getitem?mr=3246044, Zbl:1304.47061 https://zbmath.org/1304.47061, doi:10.1080/03605302.2013.864304 https://doi.org/10.1080/03605302.2013.864304, https://arxiv.org/abs/1308.3606 arXiv:1308.3606

work page doi:10.1080/03605302.2013.864304 2014

[32] [32]

Musina and A

R. Musina and A. Nazarov. On fractional laplacians -- 2. Ann. Inst. H. Poincar \' e C Anal. Non Lin \' e aire , 33(6):1667--1673, 2016. MR3569246 https://mathscinet.ams.org/mathscinet-getitem?mr=3569246, Zbl:1358.47030 https://zbmath.org/1358.47030, doi:10.1016/j.anihpc.2015.08.001 https://doi.org/10.1016/j.anihpc.2015.08.001, https://arxiv.org/abs/1408.3...

work page doi:10.1016/j.anihpc.2015.08.001 2016

[33] [33]

Nochetto, E

R. Nochetto, E. Ot \' a rola, and A. Salgado. A P D E approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. , 15(3):733--791, 2015. MR3348172 https://mathscinet.ams.org/mathscinet-getitem?mr=3348172, Zbl:1347.65178 https://zbmath.org/1347.65178, doi:10.1007/s10208-014-9208-x https://doi.org/10.1007/s10208-014-9...

work page doi:10.1007/s10208-014-9208-x 2015

[34] [34]

Embedding the braid group in mapping class groups , JOURNAL =

X. Ros-Oton. Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. , 60(1):3--26, 2016. MR3447732 https://mathscinet.ams.org/mathscinet-getitem?mr=3447732, doi:10.5565/PUBLMAT\_60116\_01 https://doi.org/10.5565/PUBLMAT_60116_01, https://arxiv.org/abs/1504.04099 arXiv:1504.04099

work page doi:10.5565/publmat 2016

[35] [35]

Bonnet-BenDhia, L

A. R \" u land. Unique continuation for fractional S chr \"o dinger equations with rough potentials. Comm. Partial Differential Equations , 40(1):77--114, 2015. MR3268922 https://mathscinet.ams.org/mathscinet/article?mr=3268922, Zbl:1316.35292 https://zbmath.org/1316.35292, doi:10.1080/03605302.2014.905594 https://doi.org/10.1080/03605302.2014.905594, htt...

work page doi:10.1080/03605302.2014.905594 2015

[36] [36]

R\" u land and M

A. R\" u land and M. Salo. Exponential instability in the fractional C alder\' o n problem. Inverse Problems , 34(4):045003, 2018. MR3774704 https://mathscinet.ams.org/mathscinet/article?mr=3774704, Zbl:1516.35535 https://zbmath.org/1516.35535, doi:10.1088/1361-6420/aaac5a https://doi.org/10.1088/1361-6420/aaac5a, https://arxiv.org/abs/1711.04799 arXiv:1711.04799

work page doi:10.1088/1361-6420/aaac5a 2018

[37] [37]

R\"uland and M

A. R\"uland and M. Salo. The fractional C alder\'on problem: low regularity and stability. Nonlinear Anal. , 193:111529, 56, 2020. MR4062981 https://mathscinet.ams.org/mathscinet/article?mr=4062981, Zbl:1448.35581 https://zbmath.org/1448.35581, doi:10.1016/j.na.2019.05.010 https://doi.org/10.1016/j.na.2019.05.010, https://arxiv.org/abs/1708.06294 arXiv:1708.06294

work page doi:10.1016/j.na.2019.05.010 2020

[38] [38]

S. K. Sahoo and M. Salo. The linearized C alder\' o n problem for polyharmonic operators. J. Differential Equations , 360:407--451, 2023. MR4562046 https://mathscinet.ams.org/mathscinet/article?mr=4562046, Zbl:1512.35678 https://zbmath.org/1512.35678, doi:10.1016/j.jde.2023.03.017 https://doi.org/10.1016/j.jde.2023.03.017, https://arxiv.org/abs/2207.05803...

work page doi:10.1016/j.jde.2023.03.017 2023

[39] [39]

V. A. Sharafutdinov. Integral geometry of tensor fields . Inverse Ill-posed Probl. Ser. VSP, Utrecht, 1994. Transl. from the Russian. MR1374572 https://mathscinet.ams.org/mathscinet/article?mr=1374572, Zbl:0883.53004 https://zbmath.org/0883.53004, doi:10.1515/9783110900095 https://doi.org/10.1515/9783110900095

work page doi:10.1515/9783110900095 1994

[40] [40]

Sylvester and G

J. Sylvester and G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) , 125(1):153--169, 1987. MR0873380 https://mathscinet.ams.org/mathscinet-getitem?mr=873380, Zbl:0625.35078 https://zbmath.org/0625.35078, doi:10.2307/1971291 https://doi.org/10.2307/1971291

work page doi:10.2307/1971291 1987

[41] [41]

G. Uhlmann. Electrical impedance tomography and C alder\' o n's problem. Inverse Problems , 25(12), 2009. Paper No. 123011, 39 pages. MR3460047 https://mathscinet.ams.org/mathscinet/article?mr=3460047, Zbl:1181.35339 https://zbmath.org/1181.35339, doi:10.1088/0266-5611/25/12/123011 https://doi.org/10.1088/0266-5611/25/12/123011

work page doi:10.1088/0266-5611/25/12/123011 2009

[42] [42]

G. Uhlmann. 30 years of C alder\' o n's problem. In S\' e minaire L aurent S chwartz---\' E quations aux d\' e riv\' e es partielles et applications. A nn\' e e 2012--2013 , S\' e min. \' E qu. D\' e riv. Partielles, pages Exp. No. XIII, 25. \' E cole Polytech., Palaiseau, 2014. MR3381003 https://mathscinet.ams.org/mathscinet/article?mr=3381003, Zbl:1331....

work page doi:10.5802/slsedp.40 2012

[43] [43]

G. Uhlmann. Complex geometrical optics and C alder \' o n's problem. In Control and inverse problems for partial differential equations , volume 22 of Ser. Contemp. Appl. Math. CAM , pages 107--169. Higher Ed. Press, Beijing, 2019. MR3965132 https://mathscinet.ams.org/mathscinet-getitem?mr=3965132, Zbl:1428.78005 https://zbmath.org/1428.78005, https://doi...

work page doi:10.1142/9789813276154_0003 2019