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Transactions of the American Mathematical Society

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Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal{R}^3$


Authors: J. Cheng, Y. C. Hon and M. Yamamoto
Journal: Trans. Amer. Math. Soc. 353 (2001), 4123-4138
MSC (1991): Primary 35R30, 31B20
Published electronically: June 6, 2001
MathSciNet review: 1837223
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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.


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  • 1. Giovanni Alessandrini, Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 3, 497–516. MR 1226614, 10.1017/S0308210500025853
  • 2. 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
  • 3. 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, 10.1088/0266-5611/12/5/003
  • 4. E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data, SIAM J. Math. Anal. 30 (1999), no. 1, 220–232. MR 1656995, 10.1137/S0036141097325733
  • 5. 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, 10.1088/0266-5611/15/4/312
  • 6. 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, 10.1016/B978-008043319-6/50038-8
  • 7. 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.
  • 8. 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, 10.1515/jiip.1998.6.4.319
  • 9. Jin Cheng and Masahiro Yamamoto, Unique continuation on a line for harmonic functions, Inverse Problems 14 (1998), no. 4, 869–882. MR 1642532, 10.1088/0266-5611/14/4/007
  • 10. Avner Friedman and Michael Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), no. 3, 527–556. MR 1017323, 10.1512/iumj.1989.38.38025
  • 11. T. S. Angell and R. Kress, 𝐿²-boundary integral equations for the Robin problem, Math. Methods Appl. Sci. 6 (1984), no. 3, 345–352. MR 761497, 10.1002/mma.1670060121
  • 12. V. Isakov, Stability estimates for obstacles in inverse scattering. J. of Computational and Applied Math. V.42 (1992), 79-88. CMP 93:01
  • 13. 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
  • 14. 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.
  • 15. Oliver Dimon Kellogg, Foundations of potential theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. MR 0222317
  • 16. 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
  • 17. M.M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics. (English translation) Springer-Verlag, Berlin (1967).
  • 18. 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.
  • 19. 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.
  • 20. Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580
  • 21. L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960), 35–45 (1960). MR 0110875
  • 22. Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
  • 23. 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, 10.1515/jiip.1994.2.3.269
  • 24. 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. CMP 2001:05
  • 25. Luca Rondi, Optimal stability estimates for the determination of defects by electrostatic measurements, Inverse Problems 15 (1999), no. 5, 1193–1212. MR 1715359, 10.1088/0266-5611/15/5/306
  • 26. R. Siegel, Boundary perturbation method for free boundary problem in convectively cooled continuous casting. Trans. ASME. Sec.C, V.108-1 (1986), 230-235.
  • 27. Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR 0455365

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Additional Information

J. Cheng
Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, China & Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376-8515, Japan
Email: jcheng@math.sci.gunma-u.ac.jp and jcheng@fudan.edu.cn

Y. C. Hon
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Email: maychon@cityu.edu.hk

M. Yamamoto
Affiliation: Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
Email: myama@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-01-02758-1
Keywords: Determination of unknown boundary, conditional stability estimation, non-smooth boundary
Received by editor(s): July 27, 1999
Received by editor(s) in revised form: June 16, 2000
Published electronically: June 6, 2001
Additional Notes: The first author is partly supported by NSF of China (No.19971016). This work was also partially supported by the Research Grants Council of the Hong Kong SAR,China (Grant numbers #9040428) and the Sanwa Systems Development Company Limited (Tokyo, Japan).
Article copyright: © Copyright 2001 American Mathematical Society