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A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation
| Title: | A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation |
| Authors: | Misu, Kuniyasu Browse this author |
| Keywords: | Viscosity solutions | | Mean curvature flow equation | | Asymptotic shape | | Large time behavior | | Deterministic game |
| Issue Date: | 1-Feb-2023 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 1149 |
| Start Page: | 1 |
| End Page: | 40 |
| Abstract: | We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, which we extend from that of Kohn and Serfaty [13]. The paper [13] gives a deterministic twoperson 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 the properties of the game trajectories given by very elementary game strategies and consider reachability of each player. Also, when the equation has driving force, we present several examples of computation of the asymptotic behavior, including a problem dealt in [8]. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/87822 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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