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

TL;DR

This paper investigates the mathematical well-posedness of a Hele-Shaw-like system derived from a shape inverse problem, and proposes an efficient numerical approach for the associated shape optimization reformulation.

Contribution

It provides a rigorous analysis of existence, uniqueness, and stability of solutions for the Hele-Shaw-like system and introduces a simple numerical method for the shape inverse problem.

Findings

Proved well-posedness of the Hele-Shaw-like system.

Developed an efficient numerical approach for shape optimization.

Established continuous dependence of solutions on data.

Abstract

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).