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

Authors:
J. Cheng, Y. C. Hon and M. Yamamoto

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

MSC (1991):
Primary 35R30, 31B20

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

Published electronically:
June 6, 2001

MathSciNet review:
1837223

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**G. Alessandrini, Stable determination of a crack from boundary measurements. Proc. R. Soc. Edinburgh V.123A (1993), 497-516. MR**94h:35259****2.**S. Andrieux, A.B. Abda & M. Jaoua, Identifiabilité de frontière inaccessible par des mesures de surface. C. R. Acad. Sci. Paris Sér. I Math. V.316 (1993), 429-434. MR**94b:35292****3.**N.D. Aparicio & M.K. Pidcock, The boundary inverse problem for the Laplace equation in two dimensions. Inverse Problems V.12 (1996), 565-577. MR**97h:35231****4.**E. Beretta & S. Vessella, Stable determination of boundaries from Cauchy data. SIAM J. Math. Anal. V.30 (1998), 220-232. MR**99j:35229****5.**A.L. Bukhgeim, J. Cheng & M. Yamamoto, Stability for an inverse boundary problem of determining a part of a boundary. Inverse Problems V.15 (1999), 1021-1032. MR**2000e:35235****6.**A.L. Bukhgeim, J. Cheng & M. Yamamoto, Uniqueness and stability for an inverse problem of determining a part of boundary. Inverse Problems in Engineering Mechanics (Nagano, 1998), 327-336, Elsevier, Oxford, 1998. MR**99m:35259****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 & M. Yamamoto, Stability in line unique continuation of harmonic functions: general dimensions. J. Inverse and Ill-posed Problems V.6 (1998), 319-326. MR**2000b:35034****9.**J. Cheng & M. Yamamoto, Unique continuation on a line for harmonic functions. Inverse Problems V.14 (1998), 869-882. MR**99d:35029****10.**A. Friedman & M. Vogelius, Determining cracks by boundary measurements. Indiana Math. J. V.38 (1989), 527-556. MR**91b:35109****11.**D. Gilbarg & N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Berlin, New York, Springer-Verlag, 1983. MR**86e:35035****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.**V. Isakov, New stability results for soft obstacles in inverse scattering. Inverse Problems V.9 (1993), 535-543. MR**94k:35189****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.**O.D. Kellogg, Foundations of Potential Theory. Dover Publications, Inc., New York (1953). MR**36:5369****16.**E. Landis, Some problems of the qualitative theory of second order elliptic equations. Russian Math. Surveys V.18 (1963), 1-62. MR**27:435****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.**S. Mizohata, The Theory of Partial Differential Equations. Cambridge University Press, London (1973). MR**58:29033****21.**L.E. Payne, Bounds in the Cauchy problem for Laplace's equation. Arch. Rational Mech. Anal. V.5 (1960), 35-45. MR**22:1743****22.**M.H. Protter & H.F. Weinberger, Maximum Principles in Differential Equations. Englewood Cliffes, New Jersey (1967). MR**36:2935****23.**A.G. Ramm, Stability of the solution to inverse obstacle scattering problem. J. Inverse and Ill-posed Problems V.2 (1994), 269-275. MR**95f:35092****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.**L. Rondi, Optimal stability estimates for the determination of defects by electrostatic measurements. Inverse Problems V. 15 (1999), 1193-1212. MR**2000k:78026****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.**A.N. Tikhonov & V.Y. Arsenin, Solutions of Ill-posed Problems. English Translation. Winston & Sons, Washington (1977). MR**56:13604**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35R30,
31B20

Retrieve articles in all journals with MSC (1991): 35R30, 31B20

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