A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910349411745792 |
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| author | Misu, Kuniyasu |
| author_facet | Misu, Kuniyasu |
| contents | We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, to which we extend that of Kohn and Serfaty (2006). Kohn and Serfaty (2006) give a deterministic two-person zero-sum game whose value functions approximate the solution to the level set mean curvature flow equation without obstacle functions. We prove that moving curves governed by the mean curvature flow converge in time to the boundary of the convex hull of obstacles under some assumptions on the initial curves and obstacles. Convexity of the initial set, as well as smoothness of the initial curves and obstacles, are not needed. In these proofs, we utilize properties of the game trajectories given by very elementary game strategies and consider reachability of each player. Also, when the equation has a driving force term, we present several examples of the asymptotic behavior, including a problem dealt in Giga, Mitake and Tran (2016). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation Misu, Kuniyasu Analysis of PDEs 35B40, 35D40, 91A50 We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, to which we extend that of Kohn and Serfaty (2006). Kohn and Serfaty (2006) give a deterministic two-person zero-sum game whose value functions approximate the solution to the level set mean curvature flow equation without obstacle functions. We prove that moving curves governed by the mean curvature flow converge in time to the boundary of the convex hull of obstacles under some assumptions on the initial curves and obstacles. Convexity of the initial set, as well as smoothness of the initial curves and obstacles, are not needed. In these proofs, we utilize properties of the game trajectories given by very elementary game strategies and consider reachability of each player. Also, when the equation has a driving force term, we present several examples of the asymptotic behavior, including a problem dealt in Giga, Mitake and Tran (2016). |
| title | A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation |
| topic | Analysis of PDEs 35B40, 35D40, 91A50 |
| url | https://arxiv.org/abs/2403.00358 |