Identifiability problems of defects with the Robin condition

Abstract

We consider the inverse problem of recovering the shape of a cavity or of a crack contained in a connected domain , and the problem of reconstructing part of the boundary ∂ 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.

Simultaneous numerical determination of a corroded boundary and its admittance

“…Even when α is known, one set of Cauchy data (2.1) and (2.3) may not be enough to determine uniquely the corroded boundary Γ 2 , as shown by the counterexamples given in [7,9,35] and some thorough numerical investigation reported in [15]. However, it turns out that two linearly independent boundary data f 1 and f 2 , one of which is positive, inducing, via (2.3), two corresponding flux measurements g 1 and g 2 , are sufficient to provide a unique solution for the pair (Γ 2 , α), [1,34,35]. The stability issue has also been recently addressed in [36] and numerical results based either on a potential approach or on a Green's integral formulation have been reported in [10].…”

Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition

“…The following counter example, inspired by the example in [10] which also appears in [19], illustrates the non-uniqueness issues for the above inverse problems using two Cauchy pairs. Let Ω be the annulus bounded by We see that b in fact depends on the combined expression µn 2 + λρ 2 and thus for fixed ρ < R (i.e., for given boundary Γ c ) and fixed n ∈ N two Cauchy pairs corresponding to the real and imaginary part of u given by (3.1) and (3.2) on the circle Γ m of radius R, provide more than one solution for µ and λ.…”

“…The uniqueness of the boundary Γ c with finitely many Cauchy pairs is an open problem, even if it is assumed that µ and λ are known. For the case of the Robin condition, i.e., µ = 0, it was shown in [1,19] that two Cauchy pairs (f 1 , g 1 ) and (f 2 , g 2 ) such that f 1 and f 2 are linearly independent and f 1 > 0 uniquely determine both Γ c and λ. Similar result can be also stated for the unique determination of µ and Γ c if λ = 0, since in this case the problem for the conjugate harmonic of u becomes a Robin problem with impedance 1/µ (see [10]).…”

A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

“…This result is optimal as well (see [8]). Let us finally mention [19], where a uniqueness result under weaker regularity assumptions on the boundary has been obtained.…”