A problem in electrical prospection and an 𝑛-dimensional Borg-Levinson theorem

Chanillo, Sagun

doi:10.1090/S0002-9939-1990-0998731-1

A problem in electrical prospection and an $n$-dimensional Borg-Levinson theorem
HTML articles powered by AMS MathViewer

by Sagun Chanillo PDF

Proc. Amer. Math. Soc. 108 (1990), 761-767 Request permission

Abstract:

We show that the Dirichlet to Neumann map for $- \Delta u + \upsilon u = 0$, determines the potential $\upsilon (x)$, for $\upsilon (x)$ satisfying the condition of C. Fefferman and D. Phong.

References

R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 45–70. MR 812283, DOI 10.1090/pspum/043/812283
Göran Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96 (German). MR 15185, DOI 10.1007/BF02421600
Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275
Sagun Chanillo and Eric Sawyer, Unique continuation for $\Delta +v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), no. 1, 275–300. MR 958886, DOI 10.1090/S0002-9947-1990-0958886-6
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, DOI 10.1080/03605308508820401
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309–360 (Russian). MR 0045281
R. G. Novikov and G. M. Khenkin, The $\overline \partial$-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 42 (1987), no. 3(255), 93–152, 255 (Russian). MR 896879
C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
Robert Kohn and Michael Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), no. 3, 289–298. MR 739921, DOI 10.1002/cpa.3160370302
Norman Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B 1949 (1949), 25–30. MR 32067
Adrian Nachman, John Sylvester, and Gunther Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys. 115 (1988), no. 4, 595–605. MR 933457, DOI 10.1007/BF01224129
John Sylvester and Gunther Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), no. 1, 91–112. MR 820341, DOI 10.1002/cpa.3160390106
John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291