The aim of this paper is to study the shape optimization method for solving the Bernoulli free boundary problem, a well-known ill-posed problem that seeks the unknown free boundary through Cauchy data. Different formulations have been proposed in the literature that differ in the choice of the objective functional. Specifically, it was shown respectively in [14] and [16] that tracking Neumann data is well-posed but tracking Dirichlet data is not. In this paper we propose a new well-posed objective functional that tracks Dirichlet data at the free boundary. By calculating the Euler derivative and the shape Hessian of the objective functional we show that the new formulation is well-posed, i.e., the shape Hessian is coercive at the minimizers. The coercivity of the shape Hessian may ensure the existence of optimal solutions for the nonlinear Ritz-Galerkin approximation method and its convergence, thus is crucial for the formulation. As a summary, we conclude that tracking Dirichlet or Neumann data in its energy norm is not sufficient, but tracking it in a half an order higher norm will be well-posed. To support our theoretical results we carry out extensive numerical experiments.

翻译：本文旨在研究求解伯努利自由边界问题的形状优化方法，这是一个通过柯西数据寻求未知自由边界的经典不适定问题。现有文献提出了多种基于不同目标泛函选取的数学表述。具体而言，文献[14]和[16]分别证明追踪诺伊曼数据是适定的，但追踪狄利克雷数据却是不适定的。本文提出一种新的适定目标泛函，用于追踪自由边界上的狄利克雷数据。通过计算目标泛函的欧拉导数与形状黑塞矩阵，我们证明新表述具有适定性，即形状黑塞矩阵在极小点处满足强制性条件。形状黑塞矩阵的强制性有助于确保非线性Ritz-Galerkin逼近方法最优解的存在性及其收敛性，因此对该表述至关重要。总结而言，我们得出结论：在能量范数下追踪狄利克雷或诺伊曼数据均不足以保证适定性，但采用高半阶范数进行追踪则能获得适定结果。为支撑理论结论，我们开展了大量数值实验。