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

Julius Fergy Tiongson Rabago; Masato Kimura

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arXiv · 2407.03083

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

Julius Fergy Tiongson Rabago·Masato Kimura

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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).

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Julius Fergy Tiongson Rabago, Masato Kimura. 2024-07-03. On the well-posedness of a Hele-Shaw-like system resulting from an inverse geometry problem formulated through a shape optimization setting. https://arxiv.org/abs/2407.03083

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