Stable determination of a crack from boundary measurements. (English) Zbl 0802.35151

The author considers the following problem: Find a simple smooth curve \(\sigma\) entering the elliptic problem \[ \Delta u = 0 \quad \text{ in } \quad \Omega \backslash \sigma, \qquad u = \text{const} \;\text{ on } \sigma, \qquad \partial_ \nu u = \delta_ P-\delta_ Q \quad \text{ on } \quad \partial \Omega \] where \(\Omega\) is a plane smooth domain, \(\overline \sigma \subset \Omega\), \(\delta_ P\) is the Dirac delta with the pole at \(P\), from the additional data \[ u = g \quad \text{ on } \quad \Gamma \subset \partial \Omega. \] The basic result is that when the data \(g\) from two different pairs of \(P,Q\) uniformly differ by \(\varepsilon\) from the same data for the crack \(\sigma^*\) then \[ \text{dist} (\sigma, \sigma^*) \leq C | \log | \log \varepsilon | |^{-1/4}. \] Here \(C\) depends on a-priori bounds on \(\sigma,\sigma^*\) which are natural and given explicitly. The proofs are based on stability in the Cauchy problem for elliptic equations and on an inspection of equipotential lines of \(u\) inside \(\Omega\).
This problem has important applications.
Reviewer: V.Isakov (Wichita)


35R30 Inverse problems for PDEs
74R99 Fracture and damage
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A35 Motion of charged particles
Full Text: DOI


[1] Duren, Pacific J. Math. 148 pp 251– (1991) · Zbl 0682.30015 · doi:10.2140/pjm.1991.148.251
[2] DOI: 10.1007/BF01766988 · Zbl 0731.31003 · doi:10.1007/BF01766988
[3] Alessandrini, Applied and Industrial Mathematics (1991)
[4] Walsh, The Location of Critical Points of Analytic and Harmonic Functions (1950) · Zbl 0041.04101 · doi:10.1090/coll/034
[5] DOI: 10.1016/0020-7225(91)90166-Z · Zbl 0825.73761 · doi:10.1016/0020-7225(91)90166-Z
[6] DOI: 10.1512/iumj.1989.38.38025 · Zbl 0697.35165 · doi:10.1512/iumj.1989.38.38025
[7] DOI: 10.2307/1971435 · Zbl 0675.35084 · doi:10.2307/1971435
[8] Nehari, Conformal Mapping (1952)
[9] Lavrent, Problemi Non Ben Posti in Fisica Matematica e Analisi (1983)
[10] Isakov, Inverse Source Problems (1990) · Zbl 0721.31002 · doi:10.1090/surv/034
[11] Hadamard, Lectures on Cauchy’s Problem (1952) · Zbl 0049.34805
[12] DOI: 10.2307/1971291 · Zbl 0625.35078 · doi:10.2307/1971291
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