On the well-posedness of a Hele–Shaw-like system resulting from an inverse geometry problem formulated through a shape optimization setting

The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a prototypical shape inverse problem and briefly propose a simple yet efficient numerical approach for solving the corresponding minimization problem. Second, we examine the existence, uniqueness, and continuous dependence of a classical solution to a Hele–Shaw-like system, which is derived from the continuous setting of a numerical discretization of the shape optimization reformulation for the shape inverse problem. The analysis is based on the methods developed by G. I. Bizhanova and V. A. Solonnikov in “ On Free Boundary Problems for Second Order Parabolic Equations ” (Algebra Anal. 12 (6) (2000) 98–139), and by V. A. Solonnikov in “ Lectures on Evolution Free Boundary Problems: Classical Solutions ” (Lect. Notes Math., Springer, 2003, pp. 123–175).