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Identifiability problems of defects with the Robin condition

We consider the inverse problem of recovering the shape of a cavity or of a crack contained in a connected domain Ω, and the problem of reconstructing part of the boundary ∂Ω itself, when a condition of the third kind (Robin condition) is prescribed on the defects. We prove a result of uniqueness by two measures: two different defects, with different coefficients of the Robin condition, cannot be compatible with the same two pairs of Cauchy data on the (accessible) boundary. In the case of cracks, we also prove that a single measure is sufficient if the coefficient of the Robin condition is known.