  This study revisits the problem of identifying the unknown interior Robin
boundary of a connected domain using Cauchy data from the exterior region of a
harmonic function. It investigates two shape optimization reformulations
employing least-squares boundary-data-tracking cost functionals. Firstly, it
rigorously addresses the existence of optimal shape solutions, thus filling a
gap in the literature. The argumentation utilized in the proof strategy is
contingent upon the specific formulation under consideration. Secondly, it
demonstrates the ill-posed nature of the two shape optimization formulations by
establishing the compactness of the Riesz operator associated with the
quadratic shape Hessian corresponding to each cost functional. Lastly, the
study employs multiple sets of Cauchy data to address the difficulty of
detecting concavities in the unknown boundary. Numerical experiments in two and
three dimensions illustrate the numerical procedure relying on Sobolev
gradients proposed herein.
