报告题目：On the Well-posedness of Tracking Dirichlet Data for Bernoulli Free Boundary Problems

报告时间：2023-11-06   10:00-11:00

报 告 人 ：龚伟  副研究员  中国科学院数学与系统科学研究院

报告地点：理学院东北楼三楼学术报告厅（302）

Abstract：The aim of this talk 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 in previous work that tracking Neumann data is well-posed but tracking Dirichlet data is not. In this talk 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-Galerkinapproximation 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.