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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 $$\Delta u = 0 \quad \text{ in } \quad \Omega \backslash \sigma, \qquad u = \text{const} \ \text{ on } \sigma, \qquad \partial\sb \nu u = \delta\sb P-\delta\sb Q \quad \text{ on } \quad \partial \Omega$$ where $\Omega$ is a plane smooth domain, $\overline \sigma \subset \Omega$, $\delta\sb P$ 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 $\sigma\sp*$ then $$\text{dist} (\sigma, \sigma\sp*) \le C \vert \log \vert \log \varepsilon \vert \vert\sp{-1/4}.$$ Here $C$ depends on a-priori bounds on $\sigma,\sigma\sp*$ 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.

35R30Inverse problems for PDE
74R99Fracture and damage
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
78A35Motion of charged particles
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