Conditional stability estimation for an inverse boundary problem with non-smooth boundary in ℛ³

Cheng, J.; Hon, Y.; Yamamoto, M.

doi:10.1090/S0002-9947-01-02758-1

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal {R}^3$
HTML articles powered by AMS MathViewer

by J. Cheng, Y. C. Hon and M. Yamamoto

Trans. Amer. Math. Soc. 353 (2001), 4123-4138

DOI: https://doi.org/10.1090/S0002-9947-01-02758-1

Published electronically: June 6, 2001

PDF | Request permission

Abstract:

In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in $\mathcal R^3$ by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.

References

Giovanni Alessandrini, Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 3, 497–516. MR 1226614, DOI 10.1017/S0308210500025853
Stéphane Andrieux, Amel Ben Abda, and Mohamed Jaoua, Identifiabilité de frontière inaccessible par des mesures de surface, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 5, 429–434 (French, with English and French summaries). MR 1209261
N. D. Aparicio and M. K. Pidcock, The boundary inverse problem for the Laplace equation in two dimensions, Inverse Problems 12 (1996), no. 5, 565–577. MR 1413419, DOI 10.1088/0266-5611/12/5/003
E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data, SIAM J. Math. Anal. 30 (1999), no. 1, 220–232. MR 1656995, DOI 10.1137/S0036141097325733
A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Stability for an inverse boundary problem of determining a part of a boundary, Inverse Problems 15 (1999), no. 4, 1021–1032. MR 1710604, DOI 10.1088/0266-5611/15/4/312
A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Uniqueness and stability for an inverse problem of determining a part of boundary, Inverse problems in engineering mechanics (Nagano, 1998) Elsevier, Oxford, 1998, pp. 327–336. MR 1675143, DOI 10.1016/B978-008043319-6/50038-8
A.L. Bukhgeim, J. Cheng & M. Yamamoto, On a sharp estimate in a non-destructive testing: determination of unknown boundaries. Applied Electromagnetism and Mechanics. K. Miya, M. Yamamoto and Nguyen Xuan Hung eds. JSAEM (1998), 64-75.
J. Cheng, Y. C. Hon, and M. Yamamoto, Stability in line unique continuation of harmonic functions: general dimensions, J. Inverse Ill-Posed Probl. 6 (1998), no. 4, 319–326. MR 1652109, DOI 10.1515/jiip.1998.6.4.319
Jin Cheng and Masahiro Yamamoto, Unique continuation on a line for harmonic functions, Inverse Problems 14 (1998), no. 4, 869–882. MR 1642532, DOI 10.1088/0266-5611/14/4/007
Avner Friedman and Michael Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), no. 3, 527–556. MR 1017323, DOI 10.1512/iumj.1989.38.38025
T. S. Angell and R. Kress, $L^{2}$-boundary integral equations for the Robin problem, Math. Methods Appl. Sci. 6 (1984), no. 3, 345–352. MR 761497, DOI 10.1002/mma.1670060121
V. Isakov, Stability estimates for obstacles in inverse scattering. J. of Computational and Applied Math. V.42 (1992), 79-88.
Laurent Lévi, Équations quasi linéaires du premier ordre avec contrainte unilatérale, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 12, 1133–1136 (French, with English and French summaries). MR 1257226
P.G. Kaup, F. Santosa & M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data. Inverse Problems V.12 (1996), 279–293.
Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317
E. M. Landis, Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables), Uspehi Mat. Nauk 18 (1963), no. 1 (109), 3–62 (Russian). MR 0150437
M.M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics. (English translation) Springer-Verlag, Berlin (1967).
M. McIver, Characterization of surface-breaking cracks in metal sheets by using AC electric fields. Proc. R. Soc. London A V.421 (1989), 179–194.
D.H. Micheal, R.T. Waechter & R. Collins, The measurement of surface cracks in metals by using a.c. electric fields. Proc. R. Soc. London A V.381 (1982), 139–157.
Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580
L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960), 35–45 (1960). MR 110875, DOI 10.1007/BF00252897
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
A. G. Ramm, Stability of the solution to inverse obstacle scattering problem, J. Inverse Ill-Posed Probl. 2 (1994), no. 3, 269–275. MR 1297687, DOI 10.1515/jiip.1994.2.3.269
L. Rondi, Uniqueness and stability for the determination of boundary defects by electrostatic measurements. Ref. S.I.S.S.A. 73/98/AF (July, 1998), SISSA ISAS Trieste, Italy.
Luca Rondi, Optimal stability estimates for the determination of defects by electrostatic measurements, Inverse Problems 15 (1999), no. 5, 1193–1212. MR 1715359, DOI 10.1088/0266-5611/15/5/306
R. Siegel, Boundary perturbation method for free boundary problem in convectively cooled continuous casting. Trans. ASME. Sec.C, V.108-1 (1986), 230-235.
Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 0455365