On the well-posedness of a Hele-Shaw-like system resulting from an inverse geometry problem formulated through a shape optimization setting
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| Format: | Preprint |
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2024
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| _version_ | 1866915359730171904 |
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| author | Rabago, Julius Fergy Tiongson Kimura, Masato |
| author_facet | Rabago, Julius Fergy Tiongson Kimura, Masato |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the well-posedness of a Hele-Shaw-like system resulting from an inverse geometry problem formulated through a shape optimization setting Rabago, Julius Fergy Tiongson Kimura, Masato Analysis of PDEs 35R30, 35R35, 35K55, 35S30, 76D27 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). |
| title | On the well-posedness of a Hele-Shaw-like system resulting from an inverse geometry problem formulated through a shape optimization setting |
| topic | Analysis of PDEs 35R30, 35R35, 35K55, 35S30, 76D27 |
| url | https://arxiv.org/abs/2407.03083 |