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.

翻译：本研究重新探讨了利用调和函数外部区域的Cauchy数据识别连通域内未知Robin边界的问题。研究考察了采用最小二乘边界数据追踪代价泛函的两种形状优化重构方案。首先，严格论证了最优形状解的存在性，弥补了现有文献的空白。证明策略中采用的论据取决于所考虑的具体公式。其次，通过建立与每个代价泛函对应的二次形状Hessian矩阵的Riesz算子的紧性，论证了这两种形状优化公式的不适定性。最后，本研究采用多组Cauchy数据来解决未知边界凹陷检测的困难。二维和三维数值实验展示了本文提出的基于Sobolev梯度的数值流程。