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).