Reconstruction of Unknown Interior Robin Boundary Using Multiple Boundary Measurements

This study investigates 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 proposes two shape optimization formulations employing least-squares boundary-data-tracking cost functionals and establishes the existence of optimal shape solutions. The study also demonstrates the ill-posed nature of the shape optimization formulations and employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary.