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A problem in electrical prospection and an -dimensional Borg-Levinson theorem
Author:
Sagun Chanillo
Journal:
Proc. Amer. Math. Soc. 108 (1990), 761-767
MSC:
Primary 35R30; Secondary 35P99
MathSciNet review:
998731
Full-text PDF Free Access
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Additional Information
Abstract: We show that the Dirichlet to Neumann map for , determines the potential , for satisfying the condition of C. Fefferman and D. Phong.
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Beals and R.
R. Coifman, Multidimensional inverse scatterings and nonlinear
partial differential equations, Pseudodifferential operators and
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vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 45–70.
MR 812283
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Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe.
Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math.
78 (1946), 1–96 (German). MR 0015185
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Calderón, On an inverse boundary value problem, (Rio
de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980,
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Sagun
Chanillo and Eric
Sawyer, Unique continuation for
Δ+𝑣 and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), no. 1, 275–300. MR 958886
(90f:35050), http://dx.doi.org/10.1090/S0002-9947-1990-0958886-6
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Sagun
Chanillo and Richard
L. Wheeden, 𝐿^{𝑝}-estimates for fractional
integrals and Sobolev inequalities with applications to Schrödinger
operators, Comm. Partial Differential Equations 10
(1985), no. 9, 1077–1116. MR 806256
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\𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒∂-equation
in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk
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]
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in electrical prospection, Comm. Pure Appl. Math. 39
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]
John
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Uhlmann, A global uniqueness theorem for an inverse boundary value
problem, Ann. of Math. (2) 125 (1987), no. 1,
153–169. MR
873380 (88b:35205), http://dx.doi.org/10.2307/1971291
- [BC]
- R. Beals and R. Coifman, Multidimensional inverse scatterings and non-linear partial differential equations, Proc. Sympos. Pure Math. 43 (1984), 45-70. MR 812283 (87b:35142)
- [B]
- G. Borg, Eine Umkerung der Sturm-Liouville Eigenwertarfgabe. Bestimmung def Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1-96. MR 0015185 (7:382d)
- [C]
- A. P. Calderón, On an inverse boundary value problem, Seminar on numerical analysis and its applications to continuum physics, Soc. Brasiliera de Mathematica, Rio de Janeiro (1980), 65-73. MR 590275 (81k:35160)
- [CS]
- S. Chanillo and E. Sawyer, Unique continuation for
and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. (to appear). MR 958886 (90f:35050)
- [CW]
- S. Chanillo and R. Wheeden,
estimates for fractional integrals and Sobolev inequalities with applications to Schródinger operators, Comm. Partial Differential Equations 10 (1985), 1077-1116. MR 806256 (87d:42028)
- [F]
- C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206. MR 707957 (85f:35001)
- [GL]
- I. M. Gelfand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk. SSSR Ser. Math. 15 (1961), 309-360. MR 0045281 (13:558f)
- [HN]
- G. M. Henkin and R. G. Novikov, Uspekhi Mat. Nauk. 42 (1987), 94-153. MR 896879 (89d:35043)
- [KRS]
- C. E. Kenig, A. Ruiz and C. Sogge, Sobolev inequalities and unique continuation for second order constant coefficient differential equations, Duke Math. J. 55 (1987), 329-348. MR 894584 (88d:35037)
- [KV]
- R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289-298. MR 739921 (85f:80008)
- [L]
- N. Levinson, The inverse Sturm-Liouville problem, Math. Tidsskr. B. (1949), 25-30. MR 0032067 (11:248e)
- [NSU]
- A. Nachman, J. Sylvester and G. Uhlmann, An
-dimensional Borg-Levinson theorem, Comm. Phys. Math. (to appear). MR 933457 (89g:35082)
- [SU
] - J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91-112. MR 820341 (87j:35377)
- [SU
] - -, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125 (1987), 153-169. MR 873380 (88b:35205)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-0998731-1
PII:
S 0002-9939(1990)0998731-1
Article copyright:
© Copyright 1990 American Mathematical Society
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