Conditional stability estimation for an inverse boundary problem with nonsmooth boundary in
Authors:
J. Cheng, Y. C. Hon and M. Yamamoto
Journal:
Trans. Amer. Math. Soc. 353 (2001), 41234138
MSC (1991):
Primary 35R30, 31B20
Published electronically:
June 6, 2001
MathSciNet review:
1837223
Fulltext PDF Free Access
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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 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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 G. Alessandrini, Stable determination of a crack from boundary measurements. Proc. R. Soc. Edinburgh V.123A (1993), 497516. MR 94h:35259
 2.
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 5.
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 7.
 A.L. Bukhgeim, J. Cheng & M. Yamamoto, On a sharp estimate in a nondestructive testing: determination of unknown boundaries. Applied Electromagnetism and Mechanics. K. Miya, M. Yamamoto and Nguyen Xuan Hung eds. JSAEM (1998), 6475.
 8.
 J. Cheng, Y.C. Hon & M. Yamamoto, Stability in line unique continuation of harmonic functions: general dimensions. J. Inverse and Illposed Problems V.6 (1998), 319326. MR 2000b:35034
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 P.G. Kaup, F. Santosa & M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data. Inverse Problems V.12 (1996), 279293.
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 E. Landis, Some problems of the qualitative theory of second order elliptic equations. Russian Math. Surveys V.18 (1963), 162. MR 27:435
 17.
 M.M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics. (English translation) SpringerVerlag, Berlin (1967).
 18.
 M. McIver, Characterization of surfacebreaking cracks in metal sheets by using AC electric fields. Proc. R. Soc. London A V.421 (1989), 179194.
 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), 139157.
 20.
 S. Mizohata, The Theory of Partial Differential Equations. Cambridge University Press, London (1973). MR 58:29033
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 L.E. Payne, Bounds in the Cauchy problem for Laplace's equation. Arch. Rational Mech. Anal. V.5 (1960), 3545. MR 22:1743
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 A.G. Ramm, Stability of the solution to inverse obstacle scattering problem. J. Inverse and Illposed Problems V.2 (1994), 269275. MR 95f:35092
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 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
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 L. Rondi, Optimal stability estimates for the determination of defects by electrostatic measurements. Inverse Problems V. 15 (1999), 11931212. MR 2000k:78026
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 R. Siegel, Boundary perturbation method for free boundary problem in convectively cooled continuous casting. Trans. ASME. Sec.C, V.1081 (1986), 230235.
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 A.N. Tikhonov & V.Y. Arsenin, Solutions of Illposed Problems. English Translation. Winston & Sons, Washington (1977). MR 56:13604
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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 3768515, Japan
Email:
jcheng@math.sci.gunmau.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, 381 Komaba, Meguro, Tokyo 1538914, Japan
Email:
myama@ms.utokyo.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994701027581
PII:
S 00029947(01)027581
Keywords:
Determination of unknown boundary,
conditional stability estimation,
nonsmooth 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
