关于Bernoulli自由边界问题追踪Dirichlet数据的适定性
On the well-posedness of tracking Dirichlet data for Bernoulli free boundary problems
AI总结：
针对Bernoulli自由边界问题中追踪Dirichlet数据不适定的问题，提出高半阶范数下的新适定目标泛函，通过理论证明其形状Hessian的强制性，并辅以数值实验验证。
摘要：
本文旨在研究求解Bernoulli自由边界问题的形状优化方法，该问题是一类经典的不适定问题，需通过Cauchy数据确定未知的自由边界。现有文献已提出多种不同的公式化形式，其差异在于目标泛函的选择。具体而言，文献[14]和[16]分别证明了追踪Neumann数据是适定的，但追踪Dirichlet数据不适定。本文提出一种新的适定目标泛函，用于追踪自由边界上的Dirichlet数据。通过计算该目标泛函的Euler导数和形状Hessian，我们证明新的公式化形式是适定的，即形状Hessian在极小值点处具有强制性。形状Hessian的强制性可保证非线性Ritz-Galerkin近似方法最优解的存在性及其收敛性，因此对该公式化形式至关重要。综上，我们得出结论：在能量范数下追踪Dirichlet或Neumann数据是不够的，而在高半阶的范数下进行追踪则是适定的。为验证我们的理论结果，我们开展了大量数值实验。
英文摘要：
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.
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