Conditional Stability in an Inverse Problem of Determining a Non-smooth Boundary

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1 Journal of Mathematical Analysis and Applications, 577 doi:.6jmaa , available online at on Conditional Stability in an Inverse Problem of Determining a Non-smooth Boundary A. L. Bukhgeim Sobole Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Acad. Koptyug Prospekt, Noosibirsk 639, Russia bukhgeim@math.nsc.ru J. Cheng Department of Mathematics, Fudan Uniersity, Shanghai 33, People s Republic of China jcheng@ms.fudan.edu.cn and M. Yamamoto Department of Mathematical Sciences, Uniersity of Tokyo, Komaba, Meguro, Tokyo 53, Japan myama@ms.u-tokyo.ac.jp Submitted by Charles W. Groetsch Received September, 998 An inverse problem for determining an unknown boundary is discussed. We prove conditional stability of logarithmic type for unknown Lipschitz boundaries under a priori assumptions. The proof is based on estimation of harmonic measures. Academic Press. INTRODUCTION In this paper, we discuss a problem in non-destructive testing, that is, determination of a boundary defect by a two-dimensional electrostatic Partially supported by the NSF of the People s Republic of China. Partially supported by Sanwa Systems Development Co., Ltd. Tokyo, Japan. 57-7X $35. Copyright by Academic Press All rights of reproduction in any form reserved.

2 58 BUKHGEIM, CHENG, AND YAMAMOTO field. We use a formulation by McIver 6 or Michael et al. 7. Let be a bounded domain in R that describes the shape of a material which has been damaged by corrosion, for example. We denote a defective subboundary by Ž. and we are required to determine from a suitable observation on an accessible subboundary. For this we use an electrostatic field už x, y., Ž x, y., Ž.. už x, y., Ž x, y., Ž.. už x, y. fž x, y., Ž x, y.. Ž.3. Here u u x, y represents an electric potential and f is considered as a boundary input. Our inverse problem is then to determine a curve from u x, y, x, y,. u where denotes the normal derivative on. The uniqueness in determining follows from the unique continuation property for the Laplace equation. The second theoretical topic is stability. Unfortunately, our inverse problem is severely ill-posed and we cannot expect good stability with the usual topologies, because an ill-posed Cauchy problem for the Laplace equation is involved. However, in the ill-posed problem, we can restore stability under suitable a priori information on the unknowns; such stability is called conditional stability. Our main purpose is to establish conditional stability in our non-destructive evaluation problem. Conditional stability of the Cauchy problem for the Laplace equation is treated by Lavrentiev 5 and Payne 9, for example, which is also a key for our proof. In Bukhgeim et al. 5, we have proved various types of conditional stability estimates according to the degrees of regularity of Ž as a priori information i.e., from C -curves to analytic curves.. From the practical point of view in non-destructive testing, it is more natural to estimate less regular unknown subboundaries than C -curves, and here we exclusively discuss conditional stability for Lipschitz curves, which are not considered in 5. For estimating Lipschitz subboundaries in this kind of inverse problem, we refer to Rondi, where conditional stability is proved for a more general elliptic equation with variable coefficients and Neumann boundary condition, in place of the Laplace equation. In comparison with, our result requires a regularity assumption for a potential field u on the whole domain but we do not need boundary values on the entire boundary and our result can be localized to a neighbourhood where we would like to evaluate unknown subboundaries Ž see Remark 6. in Section 6.. In fact,

3 STABILITY IN DETERMINING A BOUNDARY 59 our estimation is carried out separately inside a small cone whose base is in. For similar inverse problems, we refer to Andrieux et al., Aparicio and Pidcock 3, Beretta and Vessella, and Kaup et al. 3. For three-dimensional, we have to determine surfaces and we refer to Cheng et al. 7. Moreover, as other important shape determination problems we mention crack determination and obstacle determination in scattering. As for the former problem, we can refer to Alessandrini, Friedman and Vogelius 8, and Rondi, for example. For the latter, see Isakov, and Ramm. Our main result guarantees stability with a double logarithmic rate which is very weak. However, this type of weak conditional stability is usual in this kind of inverse problem Že.g.,,,,,,.. This paper is organized as follows: Section. Main result Section 3. Reduction of the theorem to two key lemmas Section. Proof of the first key lemma Section 5. Proof of the second key lemma Section 6. Concluding remarks. MAIN RESULT Let, be ordinate sets whose upper subboundaries are defective with corrosion. More precisely, let a b. For arbitrarily fixed,, m, and m, we set F FŽ,, m, m. F C, FŽ x.,xa; FŽ x., b x ; FŽ x. FŽ x. m x x, FŽ x. m,x, x, Ž.. Ž x, y. x, y FŽ x., Ž.. Ž x, y. x, y F Ž x. Ž.3. for F, F F. In other words, Ž x, y. y FŽ x., a x b j j, j,, are unknown upper subboundaries, and the corroded boundaries are given by Lipschitz continuous functions F and F. The a priori information F, F F means that the corrosion process makes the subbound-

4 6 BUKHGEIM, CHENG, AND YAMAMOTO aries shapes not so complicated, although the damaged boundaries can admit corners. However, our a priori information excludes cusps where the cone property breaks Že.g., Grisvard.. As the observation boundary, we take Let us assume that u, j,, satisfy j Ž x,.; a x b. Ž.. u Ž x, y., Ž x, y., Ž.5. j u Ž x, y., Ž x, y., Ž.6. j u j ujž x,. fjž x., Ž x,. gjž x., a x b. Ž.7. y By the unique continuation for harmonic functions, we note that there exists at most one u satisfying Ž.5. Ž.7. j. We are ready to state our main result. THEOREM.. Let m, m,, M, x Ž a, b. 3 be fixed, but arbitrary, and F, F F. Moreoer, we assume f x jž. m, j,, Ž.8. C u C, u j j j j M, j,,.9 j j and u m, j,. Ž.. j C Ž a, b, m. 3 Then there exist constants C and such that F F C a, b C, Ž.. log logž. where f f g g H Ž a, b. L Ž a, b.. Here the constants C and depend on m, m, m, m 3,, M, and F. We do not assume knowledge of boundary values on the whole boundary. That is, we note that we need not impose any boundary conditions on Ž. j j, and in applications it is common that we cannot know boundary values outside of j and. If we pose suitable boundary values on Ž. j j, then we can improve Ž.. to the rate of a single logarithm Že.g., Bukhgeim et al.. 5. Moreover, we can obtain the conditional stability of Holder type as far as

5 STABILITY IN DETERMINING A BOUNDARY 6 we consider a bounded set of analytic curves as s j 5. For determination of obstacles in inverse scattering, similar Holder stability is proved for analytic curves ŽIsakov.. C Ž j. Remark.. The a priori assumption u j M in.9 can be satisfied in general in a Lipschitz domain. See Grisvard j. Remark.. The conditional stability in our inverse problem depends heavily on the one in a Cauchy problem for the Laplace equation. Thus the following example for a Cauchy problem suggests that we cannot replace the a priori boundedness assumption in Ž.9. by u j CŽ j. M, j,, with the maximum norm: Let Ž x, y. x, y and nny u x, y e cos nx, n,, 3,.... Then u in and n unž x,. e n cos nx, u n n Ž x,. ne cos nx, y namely, the norm of the Cauchy data u Ž,.C, Ž u y.ž,. n n C, tends to as n. However, u Ž x,. n cos nx, x, so that u Ž,. n L Ž,. does not converge to as n. We note here that u and sup u for. n CŽ. n n C Ž. Remark.3. We note that, since the observation boundary is smooth Že.g.,. 9, we know the assumption Ž.. can be satisfied. The proof is based on the well-known conditional stability Že.g., Lavrentiev 5 and Payne 9. in the Cauchy problem for the Laplace equation and an estimate for a special harmonic measure. We modify an estimate for a similar harmonic measure which is used in Isakov,. n 3. PROOF OF THEOREM. Let F Ž x. F Ž x. take its maximum at x* a, b and without loss of generality we may assume F Ž x*. F Ž x*. F F. Ž 3.. C a, b Let D be a connected component of the line x x*. We set which includes a segment of D, D. Ž 3.. We note that D. Our main result depends on two key lemmas.

6 6 BUKHGEIM, CHENG, AND YAMAMOTO LEMMA 3.. There exist constants and C depending on F, m, and M such that C u CŽ.. Ž 3.3. logž. Remark 3.. Lemma 3. asserts conditional stability of log-order which is known for the Cauchy problem for the Laplace equation. However, for our purpose, we have to obtain such stability which is uniform in a domain whose upper boundary is given by F F admitting Lipschitz continuous functions. For completeness, we give the proof in Section. LEMMA 3.. Let u C. Then there exist constants and C depending on F, m, and M such that F F C a, b C. Ž 3.. logž. If Lemmas 3. and 3. are proved, then the proof of the main theorem is straightforward: F F C C a, b log C Ž logž.. C log logž. log C C 3. Ž 3.5. log logž.. PROOF OF LEMMA 3. Henceforth C k, denote positive constants depending only on F, m, and M. Moreover, we identify, R with i C. LEMMA.. Let c and d be positie constants such that a c d b. Then there exist constants C and which are indepen-

7 STABILITY IN DETERMINING A BOUNDARY 63 dent of y such that m u x, y u x, y C, c x d, y,.. Proof. See Payne 9. Let, D z x iy C zr, arg z and l, D. DEFINITION. z is called the harmonic measure for D and l if it satisfies, z D l,, z D,, z l. For the existence and uniqueness of the harmonic measure, we refer to Kellogg, for example. By using the same method in Friedman and Vogelius 8, we can see that C Ž D. for some Ž,.. LEMMA.. There exist constants C and C such that 5 6 Ž x. C x, x,, Ž.. 5 ž / Ž x. C6, x, R, Ž.3. x R where is the harmonic measure for D and l. Here C5 and C6 are independent of x and dependent on,, and R. Moreoer, if, then C is independent of,, and R. 3 6 Proof. We will prove the lemma on the basis of Isakov. We first prove the second inequality Ž.3.. We consider the conformal map z. z Then transforms D l to the following domain: ½ 5 S,,. Ž.. R

8 6 BUKHGEIM, CHENG, AND YAMAMOTO Ž. Moreover, w satisfies wž., S, wž., Ž D., wž., Ž l.. By the maximum principle, we have Let w satisfy ž / wž.,. Ž.5. R ž R / ž R / w,,, Ž.6. w,,.7 ž R / ž R / w Ž.,, Ž.8. w..9 Noting Ž.5. and applying the maximum principle Že.g.,. 9 to a harmonic function 5 wž. w in ½,, R R we have by.5 and wž. w, Ž.. R R w,.. Since, at, the domain satisfies the ball property, the strong maximum principle yields w,. x

9 STABILITY IN DETERMINING A BOUNDARY 65 Že.g.,. 9. Therefore, in view of Ž.., there exists a constant C7 such that namely, 7 ž R / w C,, ž / ž / 7 w C,. R R R Ž.3. The inequalities Ž.. and Ž.3. imply Ž.3.. Next let 3. In place of w, we consider w, the solution to wž.,,, wž.,, w,, wž.. Similarly, we can prove that w Ž. C7 for. Here C7 is independent of,, and R because w is. Since 3, we see that, R, R and w Ž. w Ž. for by the maximum principle. Therefore Ž.. yields 7 w C,, R R where C is independent of,, and R. 7 For the proof of., we introduce the conformal map: Then D l is transformed to Ž z. z. S, R,,

10 66 BUKHGEIM, CHENG, AND YAMAMOTO and w Ž Ž.. satisfies w, S, w Ž., or R, w Ž.,. As in the proof of Ž.3., in view of the maximum principle and the strong maximum principle, we see that w Ž. for S and Thus the proof of. is complete. C5 w Ž.,. In terms of Lemma., we can establish a conditional stability estimate for holomorphic functions in D. LEMMA.3. Suppose Ž z. is a holomorphic function in D. We assume that max Ž x.. If Ž z. x, M, z D, with some M, then we hae ž M / C5 x Ž x. M, x,, Ž. C x R 6 Ž x. M, x, R. M Proof. Without loss of generality, we may assume Consider the following function: We will prove that max Ž x.. x, WŽ z. Ž z. exp Ž z. log M Ž z. log. WŽ z. max WŽ z. M, z D. Ž.. z D, In fact, if. is not true, then there exists z D, such that WŽ z. max WŽ z. max WŽ z. M. Ž.5. zd z D,

11 STABILITY IN DETERMINING A BOUNDARY 67 Since z is in D,, there exists a small ball OŽ z. with centre z such that OŽ z. D,. From the theory of complex variables, we know that, for the harmonic function Ž z. in OŽ z., there exists a holomorphic function Ž z. such that Ž z. Ž z., z OŽ z., where z is the real part of z. We set VŽ z. Ž z. exp Ž z. log M Ž z. log, z OŽ z.. Then VŽ z. WŽ z., z OŽ z.. It is easy to verify that VŽ z. is a holomorphic function in OŽ z. and VŽ z. attains the maximum WŽ z. at an interior point z. Therefore the maximum principle for holomorphic functions and harmonic functions yields and so VŽ z. constant, z OŽ z., WŽ z. constant, z D. This contradicts Ž.5.. Thus we have Ž... From Ž.., we have Ž x. Ž x. M, x, or x, R. M By noting M, the conclusion follows from Lemma.. Now we can complete the proof of Lemma 3.. Let už x, y. u Ž x, y. u Ž x, y. and Žx, F Ž x... We fix c and d such that c a b d and c,d are sufficiently small. Since F F, we can choose m such that D, D Ž x, y. c x d, Ž.6. where we set D x, y R y F Ž x., F Ž x. y m x x. Henceforth we set z x iy and identify with C. Since už x, y. for Ž x, y. D, there exists a real harmonic function wž x, y. such that u iw is a holomorphic function in D and wž x,..

12 68 BUKHGEIM, CHENG, AND YAMAMOTO In fact, let be the conformal mapping from D to the unit disk z. Then the holomorphic function can be constructed by means of the Schwarz formula Že.g., 8, p. 8;, p.. where Ž. Ž z. už Ž.. H Ž. d iw, Ž.7. i D Ž. Ž z. Ž. / Ž. Ž x. už Ž.. w H Ž. d. ž i D Ž. Ž x. Ž. Since the Cauchy singular integral operator is a bounded operator from C D to C Ž D. Že.g., 8., it can be directly verified that Hence, by Lemma., we have Ž z. u C Ž M.. C Ž. m z C, z, z x,.8 where z is the imaginary part of z. Applying the first inequality in Lemma.3 in D, we obtain Ž. C5 F x y C m Ž y. C Ž M., y, F Ž x., Ž.9. CŽ M. where arctan m. Therefore we have Ž. C5 F x y C m už y. C Ž M., y, F Ž x.. CŽ M. Since u Ž x, F Ž x.., in terms of Ž.9., we have u Ž x, F Ž x.. as u x, F Ž x. u x, F Ž x. u x, F Ž x. where h is a small positive constant. u x, F Ž x. u x, F Ž x. h u x, F Ž x. u x, F Ž x. h u x, F Ž x. h u x, F Ž x. h / C5h C Mh CŽ M., Ž.. ž CŽ M.

13 STABILITY IN DETERMINING A BOUNDARY 69 We choose h so that it minimizes / C5h C Mh CŽ M.. ž CŽ M. Then we can complete the proof of Lemma PROOF OF LEMMA 3. First, by the maximum principle, we have už x., x D. Ž 5.. By the assumption.8 and., there exists a positive constant m such that m u Ž x, x., x, Ž 5.. where is independent of u and is dependent only on and m 3. Since u Ž z. is a harmonic function in a simply connected domain, there exists a harmonic function wž z. such that Ž z. u Ž z. iwž z. is a holomorphic function. It is easy to verify that Ž z. is a holomorphic function in and C Ž. M. 5.3 Here M is a positive constant which depends on M of Ž.9.. From Ž 5.., we have, for any real constant c, x, x ic u x, x iw x, x ic Ž. Ž. Ž.. Ž 5.. For the proof, we can assume that d F F F Ž x*. F Ž x*. C a, b and is sufficiently small. If F x* F x* 3, then the proof is finished. Therefore we may assume that m FŽ x*. FŽ x*. 3. Ž 5.5. Since F, F F and, we can take cones K, K, K, K given by K Ž x, y. y F Ž x*., y F Ž x*. m x x*, K Ž x, y. y F Ž x*., y F Ž x*. m x x*

14 7 BUKHGEIM, CHENG, AND YAMAMOTO and K x, y y F x*, F x* y m x x*, K x, y y F x*, F x* y m x x* satisfying K K and K K C Ž the complement of.. Put K K K, K K K and we see K K D. The angles of K at the vertices Ž x*, F Ž x*.. and Ž x*, F Ž x*.. are bounded away from at uniform distances by F, F F. We set L Ž x, y. x x*, F Ž x*. y F Ž x*.. We see that L K K D and u 3 už z., C8 C8 x for z x*, z FŽ x*., FŽ x*. Ž 5.6. Že.g., 9, p. 3.. Since Ž z. u Ž z. iw Ž z. is a holomorphic function, we have u w u w,. x y y x By 5.6, we have w 3 Ž z. C8. y Let c w Žx* if Ž Ž x*... and w w c. We have H y w w Ž x* iy. Ž x* it. dt y F x* H y u Ž x* it. dt. x F x* We can directly verify that u iw is a holomorphic function and 3 Ž z. C 8 z x*, z F Ž x*., F Ž x*., Ž 5.7. where C is a constant. 8

15 STABILITY IN DETERMINING A BOUNDARY 7 Let d d. As in the proof of Lemma 3., noting Ž 5.3., we apply the second inequality in Lemma.3 to the cone K where the bound Ž 5.6. is given in the inside segment F Ž x*., F Ž x*.. Thus we obtain C x Mž M / 3 ŽŽ Ž.. Ž Ž.. C6 dtf x* d F x*. 8 z x* it, t, F x*, 5.8 where, is dependent on F, m, and M. Here we note that in Lemma.3 corresponds to d, and can be a sufficiently small positive constant. Therefore we have 3. Hence, by the latter part of Lemma., the constant C6 is independent of d. 3 Let t, m in Ž Then we have C d 3 9 Ž x* it. C 8 M, Ž 5.9. M where C9 is a constant which depends on F and. By a conditional stability estimate Že.g., 6, p.., we know that there exists a constant, which depends on F and, such that 3 C9d x, Ž. t, m C8 sup x* it. M M M By 5., we have C d 9 8, Ž 5.. m C 3. Ž 5.. M M Since we can assume that Mm and MC, we see from 8 5. that logž Mm. d 3 C logž MC. log 9 8 C. log

16 7 BUKHGEIM, CHENG, AND YAMAMOTO Therefore we obtain F Ž x*. F Ž x*. d d C ž logž./ C ž logž./. Thus the proof of Lemma 3. is complete. 6. CONCLUDING REMARKS Remark 6.. Our method applies more locally than the method in Rondi. In fact, setting ½ 5 F Ž x. F Ž x. LipŽ F : d, d. sup d x x d x x for d d, we can state our main theorem in a more localized version: THEOREM 6.. Let F, F C, and F F, and let D be a connected component of, say, D Ž x, y. F Ž x. y F Ž x., d x d, with some d, d Ž,.. For, we assume Ž.5. Ž.7., Ž.8. Ž.., and where we set C Ž D. u C D C D, u j j j j M, j,, 6. j D Ž x, y. y FŽ x., d x d. j j Then there exist constants C and depending on m, m, m 3, M, d, d, LipŽ F : d, d., and LipŽ F : d, d. such that / F F C d, d C Ž 6.. ž log logž. for f f g g. H Ž a, b. L Ž a, b.

17 STABILITY IN DETERMINING A BOUNDARY 73 Remark 6.. We consider the case where an unknown subboundary is given by a Lipschitz continuous function. Thus our result does not directly cover the case where a subboundary has cusp points. Our methodology is applicable to the cusp case with modification and in a forthcoming paper we treat such a singular case. Remark 6.3. We assume the zero Dirichlet boundary conditions on unknown subboundaries. We shall discuss the case of the Neumann boundary condition and treat that case in a forthcoming paper. ACKNOWLEDGMENTS This paper was written during the first-named author s stay in Japan between October 997 and April 998; the stay was supported by the Ministry of Education, Japan, and Russian RFFI Grant The first-named author thanks Professor Yuusuke Iso ŽKyoto University. for the arrangements. The authors thank Professor G. Alessandrini ŽUniversity of Trieste, Italy. for valuable discussions, and they highly appreciate the referees valuable comments. REFERENCES. G. Alessandrini, Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh Sect. A 7 Ž 993., S. Andrieux, A. B. Abda, and M. Jaoua, Identifiabilite de frontiere ` inaccessible par des mesures de surface, C. R. Acad. Sci. Paris Ser. I Math. 36 Ž 993., N. D. Aparicio and M. K. Pidcock, The boundary inverse problem for the Laplace equation in two dimensions, Inerse Problems Ž 996., E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data, SIAM J. Math. Anal. 3 Ž 999., A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Stability for an inverse boundary problem of determining a part of boundary, Inerse Problems Ž 999., J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, Vol. 3, AddisonWesley, Reading, MA, J. Cheng, Y. C. Hon, and M. Yamamoto, Stability in line unique continuation of harmonic functions: General Dimensions, J. Inerse Ill-Posed Probl. 6 Ž 998., A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Uni. Math. J. 38 Ž 989., D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, V. Isakov, Stability estimates for obstacles in inverse scattering, J. Comput. Appl. Math. Ž 99., V. Isakov, New stability results for soft obstacles in inverse scattering, Inerse Problems 9 Ž 993., P. G. Kaup, F. Santosa, and M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data, Inerse Problems Ž 996., 7993.

18 7 BUKHGEIM, CHENG, AND YAMAMOTO. O. D. Kellogg, Foundations of Potential Theory, Dover, New York, M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics ŽEnglish translation., Springer-Verlag, Berlin, M. McIver, Characterization of surface-breaking cracks in metal sheets by using AC electric fields, Proc. Roy. Soc. London Ser. A Ž 989., D. H. Michael, R. T. Waechter, and R. Collins, The measurement of surface cracks in metals by using a.c. electric fields, Proc. Roy. Soc. London Ser. A 38 Ž 98., N. I. Muskhelishvili, Singular Integral Equations, Ž English translation., Dover, New York, L. E. Payne, Bounds in the Cauchy problem for Laplace s equation, Arch. Rational Mech. Anal. 5 Ž 96., A. G. Ramm, Stability of the solution to inverse obstacle scattering problem, J. Inerse Ill-Posed Probl. Ž 99., L. Rondi, Uniqueness and stability for the determination of boundary defects by electrostatic measurements, Proc. Roy. Soc. Edinburgh Sect. A., to appear.. I. N. Vekua, Generalized Analytic Functions Ž English translation., Pergamon, Oxford, 96.

Journal of Mathematical Analysis and Applications 258, Ž doi: jmaa , available online at http:

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