Determining cracks by boundary measurements.

*(English)*Zbl 0697.35165The authors study the determination of an unknown internal conductivity profile from boundary measurements of voltage potentials and corresponding current fluxes. In case one has complete boundary data it is known that an isotropic conductor is uniquely determined [see e.g. J. Sylvester and G. Uhlmann, Ann. Math., II. Ser. 125, 153- 169 (1987; Zbl 0625.35078)]. The authors consider the case of a curve \(\sigma\) of zero or infinite conductivity located inside a 2-dimensional medium with a known reference conductivity. It is shown that exactly two boundary measurements determine the presence of \(\sigma\) and its shape. Two specific boundary voltage fluxes are constructed with the property that these fluxes together with the corresponding boundary voltage potentials uniquely determine \(\sigma\). Similarly, two specific boundary voltage potentials are constructed such that they together with the corresponding boundary fluxes uniquely determine \(\sigma\). A weak stability result is also obtained.

Reviewer: P.Stefanov

##### MSC:

35R30 | Inverse problems for PDEs |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |

78A25 | Electromagnetic theory (general) |